The way of the whole space and elementary particles appear

Pair production
Of
Light velocity c with Planck’s constant =The GAP Space theory =**********************************************
Shogo Niimi
Nishigo 89, Kashima, Gamagori,
Aichi, 4430037, Japan
shogo-ni@sk2.aitai.ne.jp**********************************************
Contents
1. Outline
2. Concept and the interaction in the space(1) Find the elementary functin(2) Extension of the elementary function (a)3. Variables(1) Meaning of ω, σ and τ(2) Extension of the elementary function (b)4. Pair production of light velocity c with Planck's constant (1) Whole space H9(2) Pair production of c with (3) Collect the expressions
5. Curvature in the space(1) Curvature Φ in the whole space(2) Micro-angular δj and the motion of precession(3) Relation in each space- 1 -6. Quantum operator c-(1) Calculation of V and (2) Vector analysis (supplement)
7. Confinement and the mixture of mass
(1) Find the mass expression(2) Angular momentum and spin(3) Quantum operator G and mass
8. Unification of the interaction(1) Fractional charge and the elementary charge(2) Gravitation and the Coulomb force
9. Next stage**********************************************- 2 -1. Outline
First, I explain that this theory was built under some supposition and what
was gotten as the result.
You say that our space was born with the Big Bang. However, I do notknow that the theory explains the Big Bang itself. Also, I do not seem toexpress it in latest M-theory. Because, light velocity c and Planck's constant are handled as the axiom that are not logically found in the M-theory.
Recently, the elementary function that I led under some supposition
expressed c and . I thank for software and calculation ability of thecomputer is rising remarkably, though the way of taking variables and theway of adapting value is more important. Below, I will extract the point of
this theory.
①

I recognize the existence of whole space H where there has not adimension.

②

H had the phase transition by the exquisiteness combination of c with
.

③

H after the phase transition shift to 9-dimension space H9 that has split
into real space R3, complex space G3 and imaginary space I3, which havethree dimensions respectively.

④

Write it as follows.
phase tr

H →H 9=[ R3｜G3｜I3 ]⑤

I call G3 as the gap space. This is the unit-complex-space, which can beseen like a membrane that appeared among both as the hindering of therecombination of R3 and I3.

⑥

R3, G3, I3 do not full split, I assume that R3 and I3 are the incomplete split
which is linked by G3.

⑦

I show this status in the theoretical expression and get elementaryfunction f(θj) from there.
f (θj) = ±ℛωjexp(±σjτ)cosωjτ⑧

I discover that c and  are pair production in the elementary function.

////////////ｃ= 299792458[m1・s−1],

= 1.054571596×10−34- 3 -[J・s1]

⑨

I can assume the Sub-Bromwich=Wagner sphere in this whole space.
Therefore, curvature Φ of whole space H9 has//////////////////////Φ= 1/w.
=1.10267894×10−45⑩[s1rad−1]that radius of the sphere has w. However, you are careful that has nota metric dimension.
The spin quantum number (0, 1/4, 1/2, 1, 2, 4) is gotten from thesecondary expression what I admit that the elementary function has avector. Also, the spin angular momentum (√3 /2) of electron, and soon, is expressed there.

⑪

In the same way, I can explain why the fractional charge is not
expressed there.

⑫

I explain that the quark confinement and the mixture, also history of thenucleon and the meson that are composite particles.

⑬⑭

Explain neutrino oscillation by the technique to be same.
Actually, I have a serious question. As for it, the photon has a possibility
to have mass. If it is sure, light velocity c is not maximum speed in H9and I explain the observation fact that they have few speed differences
between the photon and the neutrino or the same.

⑮

Four interactions are unified.

- 4 -2. Concept and the interaction in the space
2-(1)Find the elementary functin

Generally, the metric space is represented in (x1, x2, x3) with 3 dimensions.
On the other hand, Minkowski introduced ‘ict’ and set space-time (x1, x2, x3,ict). I suppose the existence of imaginary space i (x1, x2, x3) here now. In
this case, this is i (ct1, ct2, ct3) rewritten. It has the extended 6 dimensions
the “real imaginary space”
C6=[ x1, x2, x3, i (ct 1, ct 2, ct 3 )]/[2 - 1]that I put this imaginary space in the Minkowski space-time. Properly,
dimension xj of the real space is replaced with ctj by the operation of light
velocity c, i.e. the photon. By the way, I cannot sense the imaginary space.
However, how about do I recognize it as the time? I can explain-i/cxj ⇒ it jori cixj ⇒ t j//[2 - 2]

that suppose ‘ic’ is an operator. Also, it composes inverse correlation to only
as the time, even if I look into the real space from the side of the imaginary
space.
By the way, I seem the time are equipped with the dimensions by [2 - 1].
Actually, you should think that it is subscript only to show three dimensionsof the imaginary space. Then, why dose the imaginary space have 3-D？
Bcause, if it is not formed, we cannot find the correlation between the real
space and the imaginary space. In other words, this is a necessary
condition.
I can rewrite relational expression [2 - 2] toc

R3 ⇔ I3[2 - 3]In this way, I am easy to take oscillation of the space.
First, I state oscillation of the electronic circuit. The amplification
circuit, the resonance circuit and the feedback circuit basically compose theoscillation circuit. I make signal input line Vin, signal output line Vout,amplification resonance circuit G(s) and feedback circuit β(s) [Fig.1].
I expresse this oscillation circuit by use the communication function, as
follows.

- 5 -/////////////////////Y(s) = G(s)・Q(s)／[1−β(s)・G(s)]/[2 - 4]

Expresse the closed loop propagator
Gf(s) = G(s)／[1−β(s)・G(s)][2 - 5]

By the way, I must make up the energy loss in the circuit for oscillate a stably.
For its purpose, I must continue to supply always there with the energy.
However, I can take output by the circuit that without the supply of theenergy continues to oscillate, ifβ(s)・G(s) = 1/[2 - 6]

is formed (Koichi Simoda, Katumi Sakurai/The basics on the electronics/Select of Physics 1/Shokabo Co.). This
circuit steadies oscillation as the constant frequency under the limitation on
the characteristic in case of the solution of ‘s’ in [2 - 6] appears as the pure
imaginary number.
Now, I operate the Laplace transform use the loop propagator. In thiscase, that you must be careful that Laplace operator ‘s’ has the complex
number generally, not the pure imaginary number. By the way, Laplace
operator ‘s’ takes limitation as the convergence of the expression. Therefore,
we must give ‘s’−ｓ= +σ+ iωor+ｓ= −σ+ iω[2 - 7]

The general circuit needs the resonance circuit to get stable oscillation
but I can remove it that does not need stable. In this case, the circuit that
does not explosive oscillation has limitation on the characteristic. The gain
has not infinity like it. Therefore, I should makeβ(s)・G(s) = −(ｓ−σ)2／ω2= 1/[2 - 8]for a solution of ‘s’ has in the right of [2 - 7].

Above, I could have done thepreparation to get the oscillation of the electronic circuit.
Then, I deliberate that use the block diagram of [Fig.2] to get a solution
about oscillation of the space. I make Vin that enters from β(s) to G(s) andmake Vout that enters from G(s) to β(s). In this case, we will call the fullclosed loop propagator of expression [2 - 5].

- 6 -First, I will define G(s) and β (s). I replace G(s) that is the samestructure of the real space with X2. Also, replace β(s) that is the samestructure of the imaginary space with (iX)2. However, I can look β(s) only
as time that ‘ic’ drop off from the real space.

For this reason, get/////////////////////X2 = −(icT)2//[2 - 9]

because I can replace (iX)2 with (icT)2.

Above, I get relation ofβ(s) = −G(s) ／c2/[2 - 10]And it is represented asβ(s)・G(s) = −G(s) ・G(s)／c2/[2 - 11]

Also, I represent [2 - 11] toG(s) ・G(s)／c2 = (s−σ)2／ω2
use [2 - 8].

[2 - 12]

Then, I get expression
G(s) = ±c(s−σ)／ω

[2 - 13]

Now, I get
Gf(s) = ±[c(s−σ)／ω]／[1+(s−σ)2／ω2]= ±cω[(s−σ)／((s−σ)2+ω2)]
by put [2 - 8] and [2 - 13] into [2 - 5].
transform in this Gf(s).

/[2 - 14]

I operate the reverse Laplace

The result is as follows.

a = L−1[Gf(s)]= ±cωL−1 [(s−σ)／((s−σ)2+ω2)]= ±cωe−στcosωτ/[2 - 15]a = ±cωe±στcosωτ/[2 - 16]

Finally, I getthat consider a solution (the left side of [2 - 7]) of Laplace operator ‘s’. Thisis the elementary function of the GAP Space theory (GAPS).

- 7 -2-(2)

Extension of the elementary-function (a)The real-space and the imaginary space have three dimensions. Therefore,
I need modification of expression [2 - 16]. There have not more contents
than the relation between one dimension of the real space and one dimensionof the imaginry space, though. Actually, the limitation that one dimensionof the imaginary space correspond to one dimension of the real space has thelimitation that is correspondence on one by one treating propagator. I
consider the above, must express the relation of 3 pieces of the space into theelementary function. Therefore, I get
aj = ±cωjexp(±σjτ)cosωjτ

j=1,2,3/[2 - 17]

If it dose not mislead you, I simplify with omits the mark of ± in [2 - 17].
It is as follows.
aj = cωje±ρjcosδj

σjτ=ρj, ωjτ=δj/[2 - 18]

Next, I will affirm that [2 - 18] is the relational expression, which wasgotten when I look at the whole space from the real space. How do I look at
the whole space from the imaginary space? In this case, get relational
expression
i aj = i cωje±ρjcosδj[2 - 19]

Here, you remember the story that find [2 - 9].

There, I adapted it thateach squared the real space and the imaginary space to the block diagram.
In other words, I must treat the elementary function is same it when the real
space and the imaginary space interfere in direct. I say it more kindly, thereal space and the imaginary space meet in the squared condition, andtherefore, the elementary function is same. This is expressed with(aj )2+( i aj)2 = aj2−aj2= 0/[2 - 20]It has disappeared. Under present condition, whole space H separates into
the real space and the imaginary space, but they disappear immediately andhave returned original. Actually, this is the phenomenon that happens
when Laplace operator ‘s’ has the pure imaginary number. Under of the fact

- 8 -that the space of us exists, Laplace operator ‘s’ has not the pure imaginary
number. Is not any thought adequate to do me? Exactly. Actually,
Laplace operator ‘s’ has the complex number. It means that the complex
space exists in the whole space. Do I express that there has the gap, thecmplex space like a membrane, between the real space and the imaginary
space? Surely, I can catch like this that the complex number is madebetween the real number and the imaginary number. I take this in theelementary function is easy. I need only to put symbol ‘i’ of the imaginary
mark in front of σ as follows.
gaj= cωjexp(±iσjτ)cosωjτ/[2 - 21]

Subscript g on the left of aj is the symbol that shows the complex expression.
Then, we get three kinds of the elementary functions. From the above,
existing the whole space is expressed with
H 9 = [ R3｜G3｜I3 ][2 - 22]9 of the right shoulder on H mean that the whole space has 9 dimensions.
Then, G3 expresses the complex space, i.e. the GAP Space. As thediscussion above, we will express by three pieces of the space do not split
fully and lead gently with G3. As for this meaning, present whole space was
not exploded and the phase transition will express it a change successfully.
Write it asphase tr

H →H 9=[ R3｜G3｜I3 ]//[2 - 23]

Then, does phase transition in the space happen easily? No, it does not.
Probably, our space that after experiences the phenomenon of [2 - 20] manytimes born. I will discuss later in this point.
By the way, the extension with the eiementary function is remaining.
The unit dimension of aj in [2 - 17] has acceleration [m1s − 2rad1] by ω
[rad1s−1]. Therefore, I suppose get the expressions of velocity and length byrepeat the integration of aj. Indeed, I get there that integrat it about τ.
For exsample,////////////////////a±j = ±cωjexp(±σjτ)cosωjτ
leads

- 9 -

/[2 - 24]

///////v+j = ∫a+jdτ/////////// = cωjexp(+σjτ)[ωjsinωjτ+σjcosωjτ]／(σj2+ωj2)[2 - 25]//////v−j = ∫a−jdτ/////////// = cωjexp(−σjτ)[ωjsinωjτ−σjcosωjτ]／(σj2+ωj2)[2 - 26]

I omit the constant that appears with the indefinite integration.

Then,////////[ωjsinωjτ+σjcosωjτ] = sin(ωjτ+δa+)・(σj2+ωj2)1／2///////////////////////////////////////////////////////////δa+ = tan−1(+σj／ωj )//[2 - 27]///////[ωjsinωjτ−σjcosωjτ] = sin(ωjτ+δa−)・(σj2+ωj2)1／2[2 - 28]//////////////////////////////////////////////////////////δa− = tan−1 (−σj／ωj)
are formed. I gather [2 - 25] and [2 - 26] to one expression to be careful, forthere not to be confusing of ±, put [2 - 27] and [2 - 28] in there. Therefore,
velocity v±j can be represented asv±j = cωje±σjτ sin(ωjτ+δa±)／(σj2+ωj2)1／2/////////////////////////////////////////////////////////δa± = tan−1 (±σj／ωj )/[2 - 29]

Incidentally, this sine function can be shown in the cosine function. In thesame way, I can get length r± and “movement difficulty” b±. But, I do notwrite them here.
Next, I will extend the elementary function in the complex space.
the way the exponential function has

By////////////////////exp(±iσjτ) = cosσjτ±i sinσ
Actually, we cannot do the addition and the subtraction between the real
number and the imaginary number in this case. In other words, I have notproblem that the real number and the imaginary number are only shown inthe complex expression for compose the expression. On such thinking, I
integrate of gaj in [2 - 21].
//////gv+j = ∫ca+jdτ- 10 -

/////////// = cωjexp(+iσjτ)[ωjsinωjτ+iσjcosωjτ]／(−σj2+ωj2)[2 - 30]///////gv j = ∫ca jdτ−−/////////// = cωjexp(−iσjτ)[ωjsinωjτ−iσjcosωjτ]／(σj2+ωj2)
OK, I look [2 - 32] tightly.
this deeply.

(−σj2+ωj2) appeared here.

I accept just as it for now.

- 11 -/[2 - 31]

I must considere

3. Variables
3-(1)Meaning of ω, σ and τI got explession a ± j use the Laplace transform.

They have varieties ofphenomena in the Laplace transform and have each way of the transformthat suited the phenomenon respectively.
Then, those ways of thetransform are agreed with some theorem.
This transform to be discussing here will be concluded on the
Bromwich=Wagner (B=W) theorem (Naohei Yamada, Toshihiro Kunieda/Laplace transform and theoperational calculus/Applied Mathematics 10/Corona Inc.).
This theorom is composed by the B=W
circle with infinite radius ℒ. However, we do not discuss it now. We need
the B=W circle only. This circle exists on the s-plane which is shown inoperator s=(Σ, iΩ).
Also, radius ℒ is explessed in ℒ =(σ, iω). By the way, complex expression
ga j has imaginary mark ‘i’ in front ofσj.
±

And then, ga±j has 3 dimensions

which is composed of three expression. First, I must adjust dimensional
quantity. For its purpose, we need three B=W circles [Fig.B=W]. Radius ℒof this B=W circles is represented as////////////ℒ = (σj, iωj),

σj = ℒ cosθj ,ωj = ℒ sinθj/[3 - 1]I square this ℒ, it has/////////////////////ℒ2 = σj2 −ωj2,∵σj⊥iωj///[3 - 2]This is the meaning of (−σj2 +ωj2)=−(σj2−ωj2) which appeared in ga±j
[2 - 32]. It depends on this, I understand that the s-space guarantees theexistence of the gap space, and a way of adopting an imginary axis in the gap
space is not tied by the s-space, also it has not in the problem even if σj has
imaginary mark ‘i’.
Also, I make relation between σj and ωj develop into the real spaceand ℒ2=σj2+ωj2 can be formed as ℒ =(σj, ωj). I can catch as the spherethat combination of these three circles.
sphere. Therefore, I rewrite to///////////w = (σj, ωj)

I call this Sub-Bromwich=Wagner∵σj = w cosθj, ωj = w sinθj- 12 -[3 - 3]

//////////w2 = σj2 +ωj2

∵ σj⊥ ωj ,////////////////= σj2 +σk2+σn2ωj = (σk+σn) ,j≠k≠n∵ωj = (σk ,σn)[3 - 4]This is the equation of the sphere in the real space. In other words, σj andωj are formed on such relation.
By the way, I must reconsider the concept in the time for the definition of
τ. Dose the time will pure physical quantity? Surely, it is the measurethat must be to estimate the future. However, I cannot decide that the time
dimension exists in our space by the fact that cannot return to the past.
Also, we can estimate the future but cannot go to the future. In the past, we
fix persistently in our memory and in the future, we build up in the present
pile. How do I catch the moveing condition up in physics? I do notmeasure it in the passage of the time; I measure it under the variation of the
momentum. The time, in this meaning, it is the measurement that isbrings in to measure the variation of the motion. It is the story, which does
not have a dream.
If we do not have the time dimension, the discussion of here has thepossibility to become empty. It is exactly but I can excuse to have utilized itthat the measurement of estimates the future as the mathematicaltechnique successfully.
What do I must explain τ? Can I explain it is the unit elementary
quantity (unity) that appears when the quantization of the momentum?
But, it has the dimension of the time. Under present condition, we seem tocontradict a previous discussion. However, τ does not appear in thesurface of the expression, but it exists as hide in the trigonometric function
and the exponential function. The excuse is formed to have utilized time
successfully as the mathematical technique about this meaning.
Specifically, I make τ (or the minimum value of τ) Planck time tpl will beproper.

I explain this reason that has relation with radius w, later.

///////////τ= 5.390557921×10−443-(2)[ s1 ](Planck time)[3 - 5]Extension of the elementary function (b)I want to discuss about the elementary function any more.

- 13 -

I immediately

understand that the vector exists in each three dimensional space. Then, I
find that ωjτ with the elementary function has the angle that is composed
between each dimensional axis and the vector.
///////////////////////a = (a1, a2, a3)///////////////////////aj = cωje±σjτcosωjτHere, I omit super-script ± on aj.

[3 - 6]
Expression [3 - 6] is the projection of thevector on each dimension. This vector is expressed with two ingredients ofcosω j τ and sinω j τ on the plane [Fig.3]. Therefore, I represent the
ingredient of the other to [3 - 6] as//////////////////////a*j = cωje±σjτsinωjτ[3 - 7]

I make the vector that composed by one pair of aj and a*j//////////////////////Aj = (aj, a*j)
Call a*j the ghost of aj.
as[3 - 8]
Also, represent the vector which is composed by a*j/////////////////////a* = ( a*1 , a*2 , a*3 )[3 - 9]From above, I make//////////////////////A = (a, a*)[3 - 10]

By the way, I look at vj which integration aj (by presumption ωj⊥σj) has//////////////////////vj = cωje±σjτ[ωjsinωjτ±σjcosωjτ]／(σj2+ωj2)///////////////////////// = ce±σjτ[ωj2sinωjτ±ωjσjcosωjτ]／(σj2+ωj2)///////////////////////// = ce±σjτωj2sinωjτ／(σj2+ωj2)∵ωj⊥σj[3 - 11]

I make vector a that is the unit elementary vector. As for ingredient aj
on each dimensions are completely same size. I call it elementary
ingredient aj. Then, one a exists only on the diagonal line in a unit space (aunit cube). In this case, one a has angle Θe between each dimension and it.
/////////////////////Θe = cos−1 [(1／3)1／2]- 14 -

//////////////////////////= 0.9553166182[rad](54.73561032 ゜)//////////////////////Θe = δ1 =δ2 =δ3(Omits proof).

[3 - 12][3 - 13]

Actually, this is the tilt angle of electron.

- 15 -

4. Pair production of light velocity c with Planck's constant 
4-(1)Whole space H9I operate the eiementary function.

First, rewrite as follows//////////////////////ωjτ = wτsinθj,σjτ = wτcosθj[4 - 1][3 - 12] has
aj = c wsinθj exp(± wτcosθj) cos( wτsinθj)gaj////////[4 - 2]c wsinθj [ cos( wτcosθj)±i sin( wτcosθj)] cos( wτsinθj)[4 - 3]= c wsinθj exp(±i wτcosθj) cos( wτsinθj)///////// =that I substitute [4 - 1] for the elementary function.

wτ has a dimensionof angle but I estimate it some constant. For now, I prepare the value fromzero to an infinity in wτ.
Next, I want to prepare the value of ωjτ andθj. However, I cannot decide the value as ωjτ, wτ and θj because there
are in the subordination each other, though. Therefore, I calculate thevalue of θj based on that specify the value of ωjτone by one.
//////////////////////θj = sin−1 (ωjτ／wτ)[4 - 4]

By the way, it has//////////////////////θj = sin−1 (Θe／wτ)[4 - 5]from ωjτ=Θe as find in the front-chapter.

Based on this, I express on[Sheet1] values that calculate aj and gaj by my computer. Incidentally, I
ignored the value of c and independently existing w for this calculation.
I get a lot of interesting phenomena on this work sheet.
First, the real part in vecter ingredient gaj of the gap space that
increases gradually while it vibrates as θj heads for π/2 from zero reaches
value [0.577350269cw ]. It is oscillating where θj is close zero. The gap
space was born in this way [Graph1].
The imaginary part of gaj is reversed with the phase of the real part andit increases gradually while oscillation same as the real part but it
disappeares on the point of θj =π/2 [Graph2].

- 16 -

[Graph5] shows the subordination of the real part and the imaginary
part in gaj. I find that gaj draw the spiral turn around central axis θ even
if I do not make 3-D graph using the software which the graph function is
excellent.
Next, the side of e+ of real space vector ingredient aj has zero by θj=0.
However, the value of it rises approximately to the infinity only θj moves
minimal. Then, it decreases rapidly as θ j increases and converges onvalue [0.577350269 cw] in θj =π/2. I seem this phenomenon with the Big
Bang as you say [Graph3].
The side of e− in aj has zero at θj =0 is the same as e+. However, e−settles in value [0.577350269cw] without increasing rapidly then θ j isheading forπ/2 [Graph4].
Two phenomena are progressing at the same time even if I take only real
space R3 in this way. This is the point that is different from the Big Bangtheory.
The appearance of imaginary space vector ingredient iaj is the same as aj.
The Big Bang happened on the side of the imaginary space, too. But, here, I
do not call it the Big Bang. Persistently, I take these all phenomena with
one thing, and call it the phase transition of whole space H. After this, H9
was born, as you know.
4-(2)Pair production of c with 

Actually, I calculated the eiementary function that gave not only ωjτ=Θebut also various value to ωjτ in the pre-section. In this reason, I wanted
a hint that decide value of radius w in the B=W circle. Casually, I found ahint that value of light velocity c in the vicinity of wτ=48.9 while repeate
calculation many times.
I repeated trial and errors that use my computer at 1 year. If I
calculate it on a paper that suppose take 30 years over. I appreciate andsurprise that the ability of the computer was improved within several years.
Actually, the computer that I had before was useless because it was one in
the Stone Age.
Then, I search for the value of c use [4 - 2].
//////////////////////a±j = cωj e±ρj cosδj- 17 -

/////////////////////////////= c wsinθj exp(±wτcosθj) cos( wτsinθj)[4 - 2]

You pay attention to the way of taking a variable here like the pre-section.
But, here, I give ωjτ=δj from π/2 to 0, first. After that, I find the valueof θj from θj=sin−1 (ωjτ/wτ). Now I suppose, value wτhas “48.9” that I
found it before. After that, I search wτ at vicinity in “48.9”. I express thecalculation result in Case①, Case②, Case③.
We know the following value.
//////////////////ｃ= 299792458[m1s−1],//////////////////／ｃ = 3.517672202×10−43

= 1.054571596×10−34[J1s1][J1m−1s2 ]The result should be surprised as you can see.
important part of each case as follows.

I enumerate the mostCase① [Condition①] wτ= 48.87668, δj = 0.00000000000553261063π／2////////////////////////////////////θj= 1.77807×10−13//////////////[Result①]//////a+j = 299792458.2 c w,

a−j = 1.05457×10−34 c w/////////////////////////////////////a−j ／a+j = 3.51767×10−43
Case② [Condition②] wτ= 48.886016, δj = Θe////////////////////////////////////θj= 0.01954296//////////////[Result②]//////a+j = 1.90228×10＋19 c w,

a−j = 6.6916×10−24 c w/////////////////////////////////////a−j ／a+j = 3.51767×10−43
Case③ [Condition③] wτ= 48.901915,

δj = 0.999999999996476π／2////////////////////////////////////θj= 0.032126893//////////////[Result③]//////a+j = 299801584.4 c w,

a−j = 1.0546×10−34 c w/////////////////////////////////////a−j ／a+j = 3.51767×10−43
I cannot lead the result out only these values in here by the ability of mycomputer. However, it excellently found the value of light velocity c in theside of a+j. In addition to it, found the value of Planck’s constant  in theside of a−j in the identical condition. The proper-ness of the gap space- 18 -

theory is agreed by this pair production of c with .
However, they have ones that must be reconsidered. I did not considere
the value of cw in there; also value c and  are not same as the actual value
in Case②.
First, I describe c and . Actually, these “values” are only the valuethat was given under some dimension by human measurement and they are
not absolute at physically. As for the meaning, I may decide that they have
c=1, =1 and actually, you have done this operate frequently. In this way, I
suppose that c and  are merely “symbol”. Actually, ratio /c is mostimportant on the physics. Therefore, I can understand that discrepancies ofthe values in Case② are not big problems. Also, ratio a − j/a+j=/c isformed that w has whatever values. I get the limited expression as follows
that rewrite the elementary function from above.
////////a+j = cωj e+ρj cosδj,ε−j = ωj e−ρj cosδj/[4 - 6]In the above considering, I ignored the unit dimension of c and .
Therefore, I expresse the side of ‘+’ is velocity v+j, also the side of ‘−’ isangular momentum −j in [3 - 11].
///////////////////////v+j = cωj2 e+ρj cosδj／(σj2+ωj2)/[4 - 7]//////////////////////−j = ωj2e−ρj cosδj／(σj2+ωj2)/[4 - 8]

I can rewrit these expressiones from the reason as/////ωj = w sin θj,

σj = w cos θj,∴(σj2+ωj2) = w2/[4 - 9]to//////////////////////v+j = c sin2θj e+ρj cosδj/[4 - 10]/////////////////////−j =  sin2θj e−ρj cosδj/[4 - 11]

Now, we can calculate in [4 - 10], [4 - 11].
These results are in Case④, Case⑤, Case⑥ as follows.
Case④ [Condition④]wτ= 48.87668,

δj = 0.00001312066853π／2///////////////////////////////////////θj= 4.21671×10−7- 19 -

//////////////[Result④]/////////v+j = 299792458 c,

−j = 1.05457×10−34 ///////////////////////////////////////−j／v+j = 3.51767×10−43
Case⑤ [Condition⑤]wτ= 48.886015 , δj = Θe///////////////////////////////////////θj= 0.01954296//////////////[Result⑤]/////////v+j = 3.71738×10＋17 c,

−j = 1.30765×10−25 ///////////////////////////////////////−j／v+j = 3.51767×10−43
Case⑥ [Condition⑥]wτ= 48.901915,

δj = 0.999999999890292π／2///////////////////////////////////////θj= 0.032126893//////////////[Result⑥]/////////v+j = 299791839.4 c,

−j = 1.05457×10−34 ///////////////////////////////////////−j／v+j = 3.51767×10−43
Case② and Case⑤ have the condition
δj = Θe(j=1,2,3)[3 - 13]

I can rewrite [3 - 11] toℛ = [ c(+) ,(￣)][4 - 12]In this case, the elementary function is expressed as///////////f (θj ) = ℛ sin2θj e±ρj cosδj///////////////////// = ( c sin2θj e+ρj cosδj , sin2θj e−ρj cosδj )[4 - 13]

I remak only e+ρj and e−ρj, because the important one is ratio /c. In
this case, I call e+ρj the kernel of c and call e−ρj the kernel of . These
results are as follows (Case⑦, Case⑧, Case⑧).
Case⑦ [Condition⑦]wτ= 48.87668, δj =0.00000000000553261063π／2///////////////////////////////////////θj= 1.77807×10−13//////////////[Result⑦]/////////c e+ρj = 1.68606×10＋21 c, e−ρj = 5.931×10−22 ////////////////////////////////////////e−ρj／e+ρj = 3.51767×10−43- 20 -

Case⑧ [Condition⑧]wτ= 48.886015,

δj = Θe///////////////////////////////////////θj= 0.01954296//////////////[Result⑧]/////////c e+ρj = 1.68606×10＋21 c, e−ρj = 5.931×10−22 ////////////////////////////////////////e−ρj／e+ρj = 3.51767×10−43
Case⑨ [Condition⑨]wτ= 48.901915,

δj = 0.999999999890292π／2///////////////////////////////////////θj= 0.032126893//////////////[Result⑨]////////c e+ρj = 1.68606×10＋21 c, e−ρj=5.93099×10−22 ///////////////////////////////////////e−ρj／e+ρj = 3.51767×10−43
Therefore, I can lead//////////////////c e+ρj = 1.68606×10＋21 c , e−ρj = 5.931×10−22 //////////////////e−ρj／e+ρj = 3.51767×10−43e−ρj・e+ρj = 1

anywhere in condition of [0≦δj≦π／2] and [48.87668≦wτ≦48.901915].
By the way, it is insufficient that  with value is only gotten. Justly, I
must consider on the theory that leads . Therefore, I lead  by thetechnique that is the same as the elementary function found in the metricspace. But, I must replace space X that in the equivalent circuit to themoment of inertia Ψ [ Fig.4 ]. The feedback circuit has “i  t”. In thisway, the elementary function is gotten as follows.
//////////////////εj = ωje−σjτcosωjτ[4 - 14]

Then, integrated it is the angular momentum as follows.
////////////////// j = ωje−σjτ sin(ωjτ+δa−)／(σj2+ωj2)1／2///////////////////////////////////////////δa− = tan−1 (−σj／ωj )[4 - 8] is the vector expression of [4 - 15].

[4 - 15]/////////////////////−j = ωj2e−ρj cosδj／(σj2+ωj2)/[4 - 8]- 21 -

4-(3)

Collect the expressions

I collect all expressions are as follows.
The scalar expressions in real space R3.
///a＋j = cωjexp(＋σjτ)cosωjτ/[4 - 16]////v＋j = cωje＋σjτ sin(ωjτ+δa±)／(σj2+ωj2)1／2////////////////////////////////////////δa+ = tan−1 ](+σj／ωj)[4 - 17]////r＋j = cωje＋σjτ[2ωjσj sinωjτ−(σj2−ωj2)cosωjτ]／(σj2+ωj2)2= cωje＋σjτ cos(ωjτ−δr＋)／(σj2+ωj2)///////////////////////////////////////δr+= tan−1 (−2ωjσj／(σj2−ωj2))[4 - 18]////b＋j = cωje＋σjτ[ωj(ωj2−3σj2)sinωjτ＋σj(3ωj2−σj2)cosωjτ]/////////////////////////////////////////////////////////////////////////////////////////////////////／(σj2+ωj2)3///////// = cωje＋σjτ sin(ωjτ+δb＋)／(σj2+ωj2)3／2[4 - 19]///////////////////δb+= tan−1 [+(3ωj2σj−σj 3)／( ωj 3−3ωjσj2)]
Incidentally, we omitted a symbol ± before but the positive and negative
solutions exist in these expressions.
The vector expressions in real space R3. The expressions which havesuper-script “＊” are ghosts.
//////////////////////////////aj = cωj e＋ρj cosδj/////////////////////////////a*j = cωj e＋ρj sinδj/[4 - 20]///////////////////////////////vj = cωj2 e＋ρj cosδj／(σj2+ωj2)//////////////////////////////v*j = cωj2 e＋ρj sinδj／(σj2+ωj2)/[4 - 21]///////////////////////////////rj = cωj3 e＋ρj cosδj／(σj2+ωj2) 2//////////////////////////////r*j = cωj3 e＋ρj sinδj／(σj2+ωj2) 2- 22 -/[4 - 22]

///////////////////////////////bj = cωj4 e＋ρj cosδj／(σj2+ωj2) 3///////////////////////////////b*j = cωj4 e＋ρj sinδj／(σj2+ωj2) 3[4 - 23]

I omit the expressions in imaginaiy space I3 because put only imaginary
symbol mark ‘i’ to the head of these expressions in R3, here.
Below, I list the elementary functions that were gotten from theequivalent circuit of the moment of inertia.
The scalar expressions in Real-Inertia-Moment-Space RΨ3.
//////////////////ε−j = ωje−σjτcosωjτ/[4 - 24]//////////////////−j = ωje−σjτ sin(ωjτ+δa−)／(σj2+ωj2)1／2/////////////////////////////////δa− = tan−1 (−σj／ωj)/[4 - 25]//////////////////ψ−j = ωje−σjτ cos(ωjτ−δr−)／(σj2+ωj2)/////////////////////////////////δr−= tan−1 (+2ωjσj ／(σj2−ωj2)).

[4 - 26]//////////////////χ−j = ωje−σjτ sin(ωjτ+δb−)／(σj2+ωj2)3／2///////////////////////δb−= tan−1 [−(3ωj2σj−σj3)／( ωj3−3ωjσj2)]
The vector expressions in real-inertia-moment-space RΨ3.

/[4 - 27]//////////////////////εj = ωj e−ρj cosδj/////////////////////ε*j = ωj e−ρj sinδj/[4 - 28]////////////////////// j = ωj2 e−ρj cosδj／(σj2+ωj2)//////////////////////*j = ωj2 e−ρj sinδj／(σj2+ωj2)/[4 - 29]///////////////////////ψj = ωj3 e−ρj cosδj／(σj2+ωj2) 2//////////////////////ψ*j = ωj3 e−ρj sinδj／(σj2+ωj2) 2///////////////////////χj = ωj4 e−ρj cosδj／(σj2+ωj2) 3- 23 -[4 - 30]

//////////////////////χ*j = ωj4 e−ρj sinδj／(σj2+ωj2) 3[4 - 31]

I omit the expression in Imaginary-Inertia-Moment-Space IΨ3 because put
only imaginary symbol mark ‘i’ to the head of these expressions in RΨ3.
They have a lot of substances.
① The vector expression of ε j has the “moment of force”. Scalar
expression εj has “energy”.
② Vector expression  j has the “angular momentum”.
Scalarexpression  j has the “Planck’s constant”.
③ Vector expression ψ j has the “moment of inertia”. As the scalarquantity, it is unclear, now.
④ χj is unclear regrettably with either.
Actually, the elementary function in gap space G3 is same as theelementary function in Gap-Inertia-Moment-Space GΨ3 completely. In thisreason is consequential because all function expresse by one phase transition
if I may explan the equivalent circuit whatever. Therefore, I use//////////////////////ℛ = [ c(+) ,(−)][4 - 32]

These scalar expressions are as follows. However, you will be careful,that there simple expressions are in the side of e− only.
±////gℳ j= ℛωje±iσjτcosωjτ/////gℋ j =±[4 - 33]ℛωje±iσjτ[ωjsinωjτ±iσjcosωjτ]／(σj2 ωj2)////////////////////////////gℋ j =−ℛωje−iσjτ sin(ωjτ+δa−)／(σj2＋ωj2)1／2/////////////////////////////δa−= tan−1(−iσj／ωj)/////gℱ j =±ℛωje±iσjτ[(σj2+ωj2)cosωjτ i 2ωjσj sinωjτ]／(σj2 ωj2)2////////////////////////////gℱ j =−ℛωje−iσjτ cos(ωjτ−δr−)／(σj2＋ωj2)///////////////////////////δr−= tan−1 (i2ωjσj／(σj2+ωj2))/////gℬ j =±[4 - 34]/// [4 - 35]ℛωje±iσjτ[ωj(ωj2 +3σj2)sinωjτ±iσj(3ωj2 +σj2)cosωjτ]////////////////////////////////////////////////////////////////////////////////////////////////／(σj2 ωj2)3- 24 -

////////////////////////////gℬ j =−ℛωje−iσjτ sin(ωjτ+δb−)／(σj2＋ωj2)3／2//////////////////////////δb−= tan−1 [−i(3ωj2σj+σj3)／(ωj 3+3ωjσj2)]// [4 - 36]

The vector expressions are as follows.
//////////////////////gℳj= ℛωj e±iρj cosδj/*////////////////////gℳ j= ℛωj e±iσj sinδj[4 - 37]///////////////////////gℋj =ℛωj2e±iρj cosδj／(σj2 ωj2)/*//////////////////////gℋ j =ℛωj2e±iρj sinδj／(σj2 ωj2)////////////////////////gℱj =ℛωj3e±iρj cosδj／(σj2 ωj2)2/*//////////////////////gℱ j =ℛωj3e±iρj sinδj／(σj2 ωj2)2///////////////////////gℬj =ℛωj4e±iρj cosδj／(σj2 ωj2)3/*/////////////////////gℬ j =ℛωj4e±iρj sinδj／(σj2 ωj2)3/[4 - 38]

[4 - 39][4 - 40]

I omit the expressions in imaginary-inertia-moment-space IΨ3 because put
only imaginary symbol mark ‘i’ to the head of these expressions in RΨ3.
In this chapter, pair production of c with  were described by the GAPS
theory.

- 25 -

5. Curvature of the space
5-(1)Curvature Φ of the whole space

In the front-chapter, we got//////////////////////48.87668 ≦ wτ≦48.901915[ rad1 ]Specifically, it is [wτ=48.886015] in condition [σj=Θe].

[5 - 1]
Then we assume

that τ has Planck time, so radius w of the B=W circle has//////////////////////w = 9.068822878×10＋44[ rad1 s−1 ][5 - 2]

If this value is permissible mathematically as infinity category, the Laplacetransform is formed in this case on the B=W theory. Curvature Φ has//////////////////////Φ= 1／w//////////////////////////= 1.10267894×10−45[ rad−1 s1 ][5 - 3]by this w. However, the unit dimension of w has angular velocity.
Properly, this radius w that is expressed by the angle same as the radiusof the Sub-B=W sphere in the whole space. It depends, I can say that thisAngular-Space has curvature Φ. In this meaning, the angular ingredientwith the vector quantity that we gave earlier is on the curvilinear coordinate
system. Incidentally, the metric space which can be seen is on theorthogonal coordinate system but there has the interesting phenomenon by
the angular space has the curvature. For details, I will describe later.
5-(2)Micro-angular δj and the motion of precession

I assume tangent xj which is touch on the sub-B=W circle has radius w. I
make unit vector x which gose out from the point of contact touch on thiscircle and xj, also, this x and the circle make angularφ. Now, I examine thisφ. Therefore, I give them some symbols like [Fig.5]. I get//////////////////////θn = tan−1 ( cotθ)[5 - 4]///////////////////////φ’ = tan−1 [(1−cosθ)／sinθ][5 - 5]- 26 -

I make θ in the case of θ has approximately zero, get///////////////////////φ = lim(θ→0) φ’/////////////////////////////= tan−1 [(1−cos θ)／sin θ][5 - 6]θn has the complementary angle of θ. From the above, it has thephase ingredient///////////////////////δa = tan−1 (σj／ωj )/////////////////////////////////=(θn+θ=π／2 )θn[5 - 7]in scalar expression vj or so, of the elementary function.
Also,//////////////////////φ’= θ／2
is proofed in the easy calculation.

[5 - 8]I get the limitation, as follows.

//////////////////////φ  θ／2

[5 - 9](It omits proof). But, it is difficult that I evaluat proper φ as this.
Can I get this value of φ on another side? Though, I must replace in which has curvature Φ with vector x which exists there because thesub-B=W circle has curvature Φ. In this meaning, φ is the tilt angle
which is accomplished by the true straight line and . By the way, I
cannot get φ directly because the size of  is unclear. Therefore, I try
calculation once again in the value of θ. Actually, I caught eyes the valueof c by chance while was examining this. I get θ 1.77807×10−13 in[Case①] when chose θ in this process. Put it in my mind, I get//////////////////////θ  1.95417×10−16[5 - 10]

as I impose the severe condition “give θ the maximum value at a±j in[Case②] does not varies by vary θ”. Then, I get the maximum limit value
from [5 - 9].
//////////////////////φ  9.77085×10−17[5 - 11]

I develop this idea into the whole space. Properly, the vector has thecurvature on it because the whole space has the curvature. In this case, I- 27 -

make r that has a true straight line vector and make я that has thecurvature of the vector. This я has the minimal tilt angle which is limitedby θ and φ compared with r. The angular of r is δj =Θe now, so the
angular of я is expressed with//////////////////////δj = Θe+φ/[5 - 12]

However, you must be careful that this calculation Θe+φ is not a simple
addition.

I describe that δ1, δ2 and δ3 are in the subordination eachother only here.
///////////////////////cosδj = sinδn cos[sin−1(cosδk／sinδn)]/////////////////////////////////////(the cycle of the order is j→k→n)[5 - 13]
I assume that this я is spinning. Then, unit vector я is doing the
motion of precession in the turn of r at tilt φ. In case of macroscopic[Fig.6(a)] can be seen overlap я and r. Also, in the case of microscopic[Fig.6(b)] expresses the motion of precession.
point in detail.
5−(3)In the future, I describe thisRelation in each space

I examined how dose whole space H9 appear in 4−(1).

Here, examine

relation between R3, G3and I3 in detail.
I make the equivalent circuit in the metric space how I applied X2 to theamplification circuit and applied (iX)2 = (icT)2 to the feedback circuit at theelectronic circuit. Therefore, I examine the relation in each space at square///////////////rℳj= ℛωj e±ρj cosδj ,////////////////iℳj= iℛωj e±ρj cosδj

gℳj= ℛωj e±iρj cosδj[5 - 14]

I make the subscript that ‘r’ has the real, ‘g’ has the gap and ‘i’ has theimaginary space of the left on each expression. Here, get them as follows.
////////rℳj2 = ℛ2ωj2 e±2ρj cos2δj ,2
2
2
gℳj = ℛ ωje±i2ρj cos2δj/////////iℳj2 = −ℛ2ωj2 e±2ρj cos2δj[5 - 15]

The positive negative reverses of rℳj2 and iℳj2, except θj is zero.

This is the- 28 -

logical reason of the anomalous value of particle and antiparticle. I mean,
antiparticle exists in the imaginary space and does not exist regularly in thereal space.
By the way, you will pay attention to e±2ρj and e±i2ρj. In case of θj is
π／2, I get//////////////////////////re

j = 1 ,

±2ρ

ge

±i2ρ

j = 1 ,

(θj =π／2 )[5 - 16]

I can catch the 2nd phase transition point in the whole space. In this case,the real space correspond as the gap space in substance, because θj does not
become above π／2. Refer to [Sheet 1]. I say another word; the gap spaceis pasted up to the real space like a membrane. I got this result that saw
them from the real space. I can say that the imaginary space correspond as
the gap space by the positive negative reverses of rℳj2 and iℳj2 when saw themfrom the imaginary space. After this, whole space H9 became steady state.
I observe the phase transition of the whole space as pair creation ofparticle and antiparticle, at presentry. However, they are not stablebecause the gap space does not exist in there.
Then, they have pairextinction immediately. In other words, the whole space that after thephase transition disappeares before grows at almost all cases. I expressethem withphase tr/////////////////////H →H 6=[ R3｜I3 ]extinction→H[5 - 17]

I must describe another one. It is why did I discuss that use the ingredients
without the vectors for examine the relation in the space. This reason is
obvious in the way of find the elementary function. In there, they treatedevery pair of dimension. Therefore, I pick each ingredient up persistently
and discuss the relation. After that, the whole space expressed by combine
three ingredients. The vector expression has a little irregular in thisprocess. I discuss it in the following chapter.
I get the result as follows.
phase tr//////////////////////H →H 9=[ R3｜G3｜I3 ]- 29 -//[5 - 18]

6. Quantum operator c-6-(1)Calculation of V and Whole-space H9 was determined in the front-chapter. From now, we discuss
to have restricted into the real-space. Then, I extract the part of the vector
display that was set before.
//////////V = ( v , v*)

Vj = ( vj , v*j)///////////v = ( v1 , v2 , v3 )vj = c sin2θj e＋ρj cosδj//////////v* = ( v*1 , v*2 , v*3 )/v*j = c sin2θj e＋ρj cosδj[6 - 1]

And then,///////// = (  , *)/ j = (  j , *j)///////// = (  1 ,  2 ,  3 )/// j =  sin2θj e−ρj cosδj/////////* = ( *1 , *2 , *3 )*j =  sin2θj e−ρj sinδj //[6 - 2]

The others expressed in the same way.
I specifically pursue substance that these expressions have. For
example, ingredient aj is the solution which was gotten from the B=W circleof j-th and cosδj expresses a ingredient on some dimension by there but thecase of carrying in aj to real space R3, think that aj one expresses the size.
In other words, they have ingredient of vector a on the composition but
fundamentally, they have independent size. Therefore, I do not say onlycosδj is j ingredient of a. Expression//////////////////////e±ρjлj = sin2θj e±ρj cosδj//[6 - 3]

is j ingredient of a. I call//////////////////////e±ρjл*j = sin2θj e±ρj sinδja ghost in that reason.
Based on the above, I operate the scalar product of V and .

- 30 -[6 - 4]

//////////////////////V・ = ( v , v*)( ,*)/////////////////////////////////= v + v *+ v*+ v**[6 - 5]

Then, calculate this everypart.
///////v = c ( sin2θe＋ρcosδ) ( sin2θe−ρcosδ)/////////////↓← ①/////////////↓← ②j ⊥ k ⊥ n
( sin2θj e＋ρj cosδj)( sin2θk e−ρk cosδk) = 0/////////////= c [ Σ( j=1,2,3 ) sin4θj cos2δj ]/[6 - 6]//////v * = c ( sin2θe＋ρcosδ) ( sin2θe−ρsinδ)/////////////↓← ③ ( sin2θj e±ρj sinδj) = −[( sin2θk e±ρk cosδk) ,/////////////↓////////////////////////////////////////////////////////////////////////////( sin2θn e±ρn cosδk)]/////////////= −2 c [ Σ( j=1,2,3 ) sin4θj cos2δj ]///[6 - 7]//////v*= c ( sin2θe＋ρsinδ) ( sin2θe−ρcosδ)/////////////= −2 c [ Σ( j=1,2,3 ) sin4θj cos2δj ][6 - 8]/////v** = c ( sin2θe＋ρsinδ) ( sin2θe−ρsinδ)/////////////= 4 c [ Σ( j=1,2,3 ) sin4θj cos2δj ][6 - 9]I get
V・ = c [ Σ( j=1,2,3 ) sin4θj cos2δj ]that the summation from [6 - 4] to [6 - 7].
Next, I operate the vector product of V and .

/[6 - 10]

Generally, it is expressedwith
V× = ( v , v*)×( ,*)
↓← ④

V 

= 0

/[6 - 11]

- 31 -

Of course, it hasv×= 0/[6 - 12]v*×* = 0/[6 - 13]

However, the solution exists in the vector product of v and *, v* and .
Here, they havev×* = −v*×

[6 - 14]Then,v×* = ( v1 , v2 , v3 )×( *1 , *2 , *3 )= [(v2 *3−v3 *2 , v3 *1−v1 *3 , v1 *2−v2 *1) ,///////////////////////////////////////////////////////////////////////////(v 1×*1 , v2×*2 , v3×*3)]
The general solution is only part  that is not part . Why is part 
gotten? As for the reason, vj and *j are not on the same dimension to
direct. In other words, the relation between vj and *j are not parallel. Of
course, the relation between vj and *k are not orthogonal. I getл*j = −[(лkfrom ③ in [6 - 7].

, лn )][6 - 15]

Also//////////////////////v j×*j = e＋ρje−ρj(лk−лn)/////////////////////////////////// = (лk−лn)[6 - 16]

I rewrite part  based on there.
(v 1×*1 , v2×*2 , v3×*3) = c (л2−л3 , л3−л1 , л1−л2 )////////////////////////////////////////////////////////////////////////////////////////////////////////////////[6 - 17]
Clearly, [6 - 17] expresses the spin of the vector [Fig.7]. From above, I get///v×* = c [(л22−л32 , л32−л12 , л12−л22) ,//////////////////////////////////////////////////////(л2−л3 , л3−л1 , л1−л2 )][6 - 18]

However, this solution becomes zero if I adapt value simply. It does not
become zero that is in only case which there has the curvature in the anguler- 32 -

space as the description in the front-chapter and δ j is equipped withmicro-anguler δj. I rewrite part  like [3 - 2], as follows.
/////////////////////////лj2−лk2 = (лj + iлk) 2∵лj⊥лk/[6 - 19]

It is the scalar that exists in the gap space and appears in the real space. In
other words, v×* is the vector on the real space which has core of the gap
space in case of the whole space has the curvature.
I replace (лj+iлk) to (лj, iлk) easy to evaluate. The subordination of
лj and iлk have a circle O to draw in the event of the vector motions ofprecession which was expressed in [Fig.6]. I can say that (лj, iлk) motionsof precession with minimal changes of δ j.
However, this is thephenomenon that exists in the gap space only.
I think it is the property (the moment) that is the base of the “force” and“charge”. In other words, the one, which can be seen in the real space,
seems to base of the “moment of the force” and the one, which can be seen in
the gap space, seems to base of the “moment of the charge (magnetic
moment)”.
Next, I get////v*×= ( v*1 , v*2 , v*3 )×(  1 ,  2 ,  3 )///////////////= c [(л32−л22 , л12−л32 , л22−л12) ,////////////////////////////////////////////////////(л3−л2 , л1−л3 , л2−л1 )][6 - 20]from [6 - 14]. Part  reverses the area in the real space of c  to part .
Part  reverses the spin direction of the vector to part .
Incidentally, the dimension of c  becomes [V1A1m1s1] in the VAMS
dimension system (Isao Imai/The electromagnetism dimension is not difficult/KAGAKU/Iwanami Shoten/2002, 1).
Discussion of in here, I can extend it in the gap space. Express V is gV
and  is g , in there.
////////////gv・g = c [ Σ( j=1,2,3 ) sin4θj e+iρj e−iρj cos2δj ]//////////////////////↓←
e+iρj e−iρj = 1//////////////////////↓///////////////////////= c [ Σ( j=1,2,3 ) sin4θj cos2δj ]*/////////////gv×g  =c [(жj2−жk2 ) , (жj−жk)] [ j=1,2,3 k=1,2,3 ]- 33 -/[6 - 21]

//////////////////////↓←
жj =лj//////////////////////↓
жj2 =лj2///////////////////////= c [(лj2−лk2 ) , (лj−лk)] [ j=1,2,3 k=1,2,3 ]Scalar product gV・g  becomes the same as V・.
////////////////////////////////g

V・g  = V・/[6 - 22][6 - 23]

Also, I can see that the part of the property leaks into the real space on thevector product.
In this section, we found the “quantum operation c  has spin with the
motion of precession at tilt angular δj in the curvature space”.
Clearly, the cause of the anomalous magnetic moment is this motion ofprecession. Also, this motion of precession raises the fluctuation.
6-(2)Vector analysis (supplement)

Here, I state the vector operation method that appears after this.
First, I reconsider the scalar product that operated in the front-chapter.
I have ③ in [6 - 7], there.
( sin2θj e±ρj sinδj) = −[( sin2θk e±ρk cosδk) , ( sin2θn e±ρn cosδk)]
However, I can expresse( sin2θj e±ρj sinδj) = [( sin2θk e±ρk cosδk) , ( sin2θn e±ρn cosδk)]////////////////////////////////////////////////////////////////////////////////////////////////////////////////[6 - 24]by the direction and the order of the vector.
I appear them for the future, as follows.
R = (r, r*)
rj = cωj3 e＋ρj cosδj／(σj2+ωj2) 2

r*j = cωj3 e＋ρj sinδj／(σj2+ωj2) 2

rj = c яj

r*j = c я*jяj = sin3θj e＋ρj cosδjя*j = sin3θj e＋ρj sinδj /[6 - 26]

Then, I get- 34 -[6 - 25]

//////////////////////я*j = (яk , яn )from [6 - 20].

/[6 - 27]

Also,/////////////////////R・R = rr + rr* + r*r+ r*r*/////////////////////rr = 1Σ(j=1,2,3) [ rj2 ][6 - 28]/////////////////////rr* = 2Σ(j=1,2,3) [ rj2 ]//////////////////////r*r = 2Σ(j=1,2,3) [ rj2 ]//////////////////////r*r* = 4Σ(j=1,2,3) [ rj2 ][6 - 29][6 - 30][6 - 31]

Next, I explain the scalar product about vector a by fractional vector (1／b).
a・(1／b) = a・[b／(b・b)]/////////////////////////= a・b／bj2/////////////////////////= ajbj／bj2(= aj／bjespecially )///////// /[6 - 32]

Incidentally, (1／b) is not reverse vector b−1 of b. Call it the “scalar fraction
product calculation”.
Also, I explain the vector product a×(1／b).
a×(1／b) = a×[b／(b・b)]= a×(b／bj2)/

This is the “vector fraction product calculation”.

- 35 -////[6 - 33]

7. Confinement and the mixture of mass
7-(1)Find the mass expression

I think that whole-space H9 has viscosity, because light velocity is limited in c.
Generally, the kinematic viscosity that measures viscosity is a term. The
persons who believed that the space has filled with ether thought thiskinematic viscosity creates mass are the primary cause (Max Jammer/Concepts of
Mass/Kodansha).
I replace this idea to the measure of the movement difficultynow.
I integrate elementary function aj, and get the movement difficulty.
B = (b, b*)
I use “angular momentum”  and B for express mass.
value of .

|| is absolute///////////////////////｜｜ =  j//////////////////////////////////// = [Σ(j=1,2,3) sin4θje−2ρj cos2δj ] 1／2/[7 - 1]

Then I get mass expression/////////1M j =  j／( A・B )//// [7 - 2]//////////// = [ j／( c2Σ(j=1,2,3) sin5θje＋2ρj cos2δj )]//////////// = ( w／c2 )[(Σ(j=1,2,3) sin4θje−2ρj cos2δj ) 1／2////////////////////////////////////////////////////////////////////／(Σ(j=1,2,3) sin5θje＋2ρj cos2δj )]By the way, the mass expression is not only one but also seven kinds at least.
by using this  j, as follows.
////2M j =  j／( V・R )/[7 - 3]////////// = [ j／( c2Σ(j=1,2,3) sin5θje＋2ρj cos2δj )]/////3M j =  j V／( A・R・R)[7 - 4]- 36 -

////////// = ( j／c2 )[( Σ(j=1,2,3) sin5θje＋2ρj cos2δj) ／(ΣjAjRj)(ΣjRjRj)]
4M j =  j R／( V・V・B)

5M j =  j B／( R・R・R)

6M j =  j A／( V・V・V)

7M j =  j (A・A・R)／( V・V・V・V・V)(A part was simplified because it became long). Probably, the movement
difficulty becomes empty by these expressions.
In addition to the above, I can make mass expression
0M j = E j／( V・V )[7 - 5]////////////////////////// = ( E j／[ c2 (Σ(j=1,2,3) sin4θje＋2ρj cos2δj )]even if use “moment of force”//////////////////////E = (ε, ε*)[7 - 6]

as described in 4-(2). Ej has energy that is the absolute value of E. Which
one should I choose? However, I get the identical mass expression mj that
pay attention to these one dimensions only.
It is the single dimension
analysis. It has
mj =  w／( c2 sin3θje＋3ρj cosδj )//[7 - 7]

This is the Fundamental Mass Expression.
It seems to be that 0Mj to 7Mj expresse the composit particle because
fundamental mass mj exists in there. Here, I examine 2Mj as an example
but do not specify each expression. Rewrite it to
2M j = ( w／c

2 )[(Σ

−2ρ
4(j=1,2,3) sin θjejcos2δj) 1／2／(ΣjVjRj)]

Pay attention to Qj =(Σ(j=1,2,3) sin4θje−2ρj cos2δj) 1／2.

/// [7 - 8]

The details are

Qj = ( sin4θje−2ρj cos2δj +sin4θke−2ρk cos2δk/////////////////////////////////////////////////////////////+sin4θne−2ρn cos2δn ) 1／2/[7 - 9]This expression composed by three ingredients that cannot resolve. Surely,
such elementary particles exist. For example, “proton” that I cannot takecomposed by three quarks out independently and so on.
Also, I can make- 37 -

Qj = ( sin4θje−2ρj cos2δj +sin4θje−2ρj sin2δj) 1／2// [7 - 10]

by operate [7 - 5]. This is the “meson” that is composed by two quarks andso on.
I seem that “three generations” are three B=W circles on each three
dimensions and can proof the “mixture of mass and the quark confinement”
by [7 - 9], [7 - 10]. Also, I can explain that the mass expression has the“mixture of the generation” by one of mj appears one of generation
Addition, I can describe the foundation cause of the “neutrino
oscillation” by the mixture of the generation.
7-(2)Angular momentum and spin

Once, I get the mass expression, it is easy to find the other physical quantity
on the classical. However, do not make it a principal objective and I
reconsider spin. Begin at 2Mj as previous. Then, momentum 2P has//////////////////////////////////2P= 2M j V{ = j V／( V・R )}/////////////////////////// =  j／R[7 - 11]

Angular momentum 2L has///////////////////////////////2L= 2P×R{ = ( j／R) ×R}///////////////////////// = ( j／Rj2)R×R///////////////////////// = 0∵R  R

Therefore, I use r and r* for the replacement of R.

/ [7 - 12]
Refer from [6 - 24] to [6 -27], I get
2
2la = ( j／rj )r×r

////

= 0
/////////////////////////////2lb = ( j／rj

*2)r*×r*- 38 -/// [7 - 13]

//////////////////////// = 02lc = ( j／rjrj)r×r*= [1]( j／Σ(j=1,2,3) [ rj2 ]) r×r*
* */////////////////////////////2ld = ( j／rjrj )r ×r/////////////////////// = [1／2]( j／Σ(j=1,2,3) [ rj2 ]) r*×r///////////////////2le = ( j／rj*rj)r×r*//////////////////// // = [1／2]( j／Σ(j=1,2,3) [ rj2 ]) r×r*
* * */////////////////////////////2lf = ( j／rj rj )r ×r/////////////////////// = [1／4]( j／Σ(j=1,2,3) [ rj2 ]) r*×r2

s spin quantum number appeared here.
2

s = 0 , 1／4 , 1／2 , 1/// [7 - 14]

Spin quantum number [1／4] is an interesting value.
The vector quantum that exists in the quantization unit space has
elevation angle Θe that was fixed on each space dimention. Also, I describethat this elevation angule Θe is the tilt angle of electron already. Therefore///////////////////////cos2δj = 1／3/[7 - 15]

will be formed even if cos2δj of//////////////////////Σ(j=1,2,3) [ rj2 ] = (c2／w2)Σ(j=1,2,3) sin6θje−2ρj cos2δj //[7 - 16]
considers the micro angular.

I take this cos2δout. Then, get/////////////////////Σ(j=1,2,3) [ rj2 ] = (c2／w2) cos2δΣ(j=1,2,3)[ ℌj2 ]Next, I take cosδout in  j, as follows.
//////////////////////// j =  cosδ[Σ(j=1,2,3) sin4θje−2ρj ] 1／2- 39 -[7 - 17]

///////////////////////////// =  cosδ[Σ(j=1,2,3) ℏj2 ] 1／2//[7 - 18]

In other words,/////////////////////cosδ／cos2δ = √3/

[7 -19]is included on a series of expressions in angurae-momentum 2l that you cansee [7 - 13]. Considering above, I can rewrit them as follows.
22la = ( j／rj )r×r/// [7 - 20]///////////////////////// = 0*2 **////////////////////////////////2lb = ( j／rj )r ×r= 0
2lc = ( j／rjrj)r×r*= [√3][(Σ(j=1,2,3) ℏj2 ) 1／2／Σ(j=1,2,3) ℌj2]я×я** *////////////////////////////////2ld = ( j／rjrj )r ×r///////////////////////// = [√3／2] [(Σ(j=1,2,3) ℏj2 ) 1／2／Σ(j=1,2,3) ℌj2]я*×я
**
2le = ( j／rj rj)r×r= [√3／2] [(Σ(j=1,2,3) ℏj2 ) 1／2／Σ(j=1,2,3) ℌj2]я×я** * */////////////////////////////////2lf = ( j／rj rj )r ×r///////////////////////// = [√3／4] [(Σ(j=1,2,3) ℏj2 ) 1／2／Σ(j=1,2,3) ℌj2]я*×я
Spin angular momentum 2S appeared in here.
//////////////////////////////2S

= 0,

√3 ,

√3／2 ,Gather them.

√3／4 

/[7 - 21]

I consider them in the simple use the single dimension space. For
example, I have vector P in here. Suppos that this P has tilt angle Θe to- 40 -

axis z. The projection on z of this P is expressed in pj = Pj cosΘe. This isreplaced in Pj = pj／cosΘe. In other words, get Pj =√3 pj. By the way, it ispj = j／[ rj2 ] in case of now, so we get
Pj = √3  j／[ rj2 ]This vector P is expressed with
Pj = √3  j／Σ(j=1,2,3) [ rj2 ]because it has tilt angle Θ e on all axes x, y, and z.

I extended thetechniques that explain the spin quantum number of electron s and spin
angular momentum S by the inclination of the vector to be often used.
I get spin quantum number 3s and spin angular momentum 3S by pursue
from mass 3M, as follows.
////////////3

s = 0,///////////3S =

7-(3)

1／4 ,

1／2 ,

0, √3／4 ,

1,

2,√3／2 ,

4

√3 ,

/
2√3 ,[7 - 22]4√3  /[7 - 23]

Quantum operator G and mass

I got quantum operator c  in 6-(1). Give it two symbols,
Gj = V・///////G = V×

// { = c [ Σ( j=1,2,3 ) sin4θj cos2δj ]{ = reference [6 - 18], [6 - 20] }I make some mass expressions based on them.
//////////////////////////////GMj// [7 - 25]

I get mass expression= Gj／[ V・(V×R )]///////////////////////////= ／(V×R )} // /// [7 - 24]{ =・(V×R )／(Vj×Rj )2 }/// [7 - 26]///////// [7 - 27]///////////////////////////= ・(V×R )／[2 g( vj×rj )]by Gj, because the mass is a scalar. The upper and lower expression
changes a solution by the preceded calculation. Then, I get///////////////////////Gj／[ V・(V×R )] ≠ ／(V×R )- 41 -/// [7 - 28]

[7 - 27] expresses spin.
Next, I use G then get
G Mj= G／[ V・V・R ]= G・R／[ Vj 2Rj 2 ][7 - 29]

I can create the mass expression that has spin by quantum operator c , inthis way.
Quantum operator G shows the property and Gj shows size. In this way,
I seem the mass needs enormous energy by the mass expression becomescomplexity in the real world. I take the property of the quantum operator
below.
① Almost all energy is converted into the mass in case of the isolate
quantum operator cannot move around in some cause.
② The energy is distributed the minimal bond energy of elementary
particles and the huge mass in case of the unisolat quantum operator
does not move around.
③ The energy is distributed the maximal bond energy of elementary
particles, kinetic energy and the comparatively small mass in case of theunisolat quantum operator can move.
④ Almost all energy is coverted to kinetic energy and zero or minimalenergy for the mass in case of the isolated quantum operator can movearound.
I can adapt all spin to all property. Then, ④ seems to express a photon anda neutrino. I cannot exclude the possibility that the photon has theminimal mass in this meaning. If the photon has mass, the light velocity
does not maximum speed in the whole space and it agrees the proper-ness ofthe observation result that a speed difference between a photon and aneutrino is hardly seen.
Therefore, light velocity c takes the consistence with Planck's constant in this way.
By the way, the spin exists in the mass expression that is equivalent to
the fermion. I can create a lot of expression that describe it. For example,
it is expressed with- 42 -

////////////////////////////////0Mj = E／( V×V )/////////////////////////// = E・( V×V ) ／( Vj×Vj )2[7 - 30]/////////////////////////// = E・( V×V ) ／[2 f ( vj×vj )]and so on that use “moment of force” E again. Value “2” is naturally led toinside [7 - 23] and the expression in this place by the calculation. Spinquantum number s=1／2 is expressed.
Incidentally, 1Mj to 7Mj appear the same expressions in the same way.
call it the “Duality”.

I

The relation between Planck mass Mp and gravitational constant GN is/////////////////////Mp = (c ／GN) 1／2
I adjust it to the GAPS theory, as follows.
G = GN Mp2,

Gj = GN Mp2/[7 - 31]

Then, I extend them to the gap space,///////////////////////////////gG= gV×g ,

g Gj= gV・g / [7 - 32]

Therefore, get2////////////////////////////////gG = GN Mp ,//2
gGj = GN Mp[7 - 33]

The property of gG leaks into the real space and absolute value Gj andThis discussion has thegGj becomes equal, as I gotten in [6 - 21], [6 - 23].
phenomenon describe essence in whole space H9.

- 43 -

8. Unification of the interaction
8-(1)Fractional charge and the elementary charge

The Coulomb’s law is
Fj = Qj・E
It is rewritten to
Fq = Nqj 2／rj 2( N = n／4πε0 ,‘n’ has not dimension)

Here, I make////////////////////Fq = Nqj 2／( R・R)//

[8 - 1]

I suppose that rj is same as R in the GAPS theory, and ‘n’ has the reciprocal offine structure constant α here now. Then, get
Nqj 2 = c (α= e2／4πε0 c )

By the way, I necessary make vector Nqj 2 in some case from [8 - 1], then I
make that use G and Gj as follows.
qj 2 = Gj／N

[8 - 2]//////////////////////qj 2 = G／N/[8 - 3]

Therefore, [8 - 1] becomes
Fq = G／Rj 2//[8 - 4]by [8 - 3]. Write down one of this specific expression.
1fq = g／rj

2///////////////////////// = v×*／rj *2[8 - 5]Here, I think about electric charge qj.

[8 - 2] and [8 - 3] have threeelements and spin that cannot be divided as we discussed about the quantum
operator in 6-(1). In this reason, I conclude that the fractional charge
cannot exist in the real world.
Incidentally, I make that one of element is a charge with one of quark.

- 44 -

In this meaning, I can describe what the fractional charge is confined bysame as the confinement theory with mass. Therefore, in case of/////////////////////qj = ( Gj／N )1／2/////////////////////////= [ c ( Σ( j=1,2,3 )лj2 )／N ] 1／2/////////////////////////= [ c (л12 +л22 +л32)／N ] 1／2, three 1／3 fractional charges compose one electric charge.

[8 - 6]
Also, I consider//////////////////лj*2 = (лk2 , лn2 )[8 - 7], [8 - 6] is rewritten to
qj = [ c (л12 +л1*2 )／N ] 1／2//[8 - 8]

I can understand what this expression is composed by one 1／3 fractionalcharge and one 2／3 fractional charge.
8-(2)Gravitation and the Coulomb force

I explain the gravitation of the GAPS theory.
Generally, the Planck mass is given in
mp = ／( c2 tpl )I place it on my mind.

Then, show [7 - 7] and those ghosts are as follows.

//////////////////////mj =  w／( c2 sin3θje＋3ρj cosδj )//////////////////////mj* =  w／( c2 sin3θje＋3ρj sinδj )[8 - 15]

I take the limitation of this ghost [8 - 15] as follows. The phase transition
starts from θj→π／2 in δj→0. I getlim[θj→π／2 ,δj→0] mj* = ／( c2 tpl )from//////////////////////lim[θj→π／2] sin3θj = 1 ,//////////////////////lim[δj→0] sinδj =δj ,[8 - 16]lim[θj→π／2] e＋3ρj = 1
δj = wτ,τ= tplClarifying by this, the mass expression of the GAPS theory includes Planck- 45 -

mass. Also, I understand what the Planck mass appears in which situation.
The general gravitation is given in
FG = GN mj2／rj2
Of course, I can modify this geavitational constant GN into
GN = c ／mp2= ( V・)／1Mj2
and so on.

[or= ( V×)／1Mj2 ]/ [8 - 17]

Therefore, I get
FG = G／Rj2[8 - 18]

The Coulomb force [8 - 4] and this gravitation [8 - 18] are the same
completely in the given limitation. By the way, if the angular space does nothave the curvature, this force becomes zero as I described [6 - 11]. However,in case of the curvature exists, the amount of force gets feeble force,///////////////////////FG = G／Rj2////////////////////////////= ζ0/////[8 - 19]

This is the gravitation, the force that forms the present space.
The solutions with specific force are as follows.
2*////////////////////////////////1f = v× ／rj/// / [8 - 20]**
2f = v× ／rjrj/ // [8 - 21]*2*
3f = v× ／rj

Incidentally,/////////////////////v×*=×v*,
Also,/////////////////////v×*=−*×v,/// [8 - 22]v*×=*×v
v*×=−×v*

expresses spin-flip. I suppose what these forces become different one eachother so the spin quantum number to be decided by rj2, rjrj*, rj*2. Also, I
seem that “v×”, “v*×*” are zero even if the angular space has curvature.
Then, I can assume that the negative force exists in all of them.

- 46 -

Actually,the general solution has 18 kinds of force. But they are settled into 3 kinds
by degeneracy. I understand that these forces are the same ones at the
point of the phase transition in the whole space or in the situation that cantake the limitation at the present space.
I extract all kinds of the force to avoid misunderstanding. We have fourkinds of the force, “1f”, “2f”, “3f” and amount of force “FG” includes them.
Here, I could prove the Grand Unified Theory.

- 47 -

9. Next StageWe sow that all pending issues are solved in the GAPS theory. However, theunspecific solution exists there. For example, we could not find the value ofthe quark-mixing angle and so on logically. You will say that I selectedvalue with the electric charge and the gravitation constant only. I did not
make relation between the anomalous magnetic moment and the curvatureof the angular space clear. However, I think that we have the solvent ofthese issues. I will show an equivalent circuit like [Fig.8] as one guidepost.
I can bring this equivalent circuit in the whole space that is made by theoscillation. However, I do not tell the much, because they have possibility tolose sight of the whole by this very complicated discussion. Also, the reason
why I do not step into this place more than it is in Nobel Prize winning
performance by Doctor Masayoshi Koshiba. Say,
“The neutrino oscillation will be proved now; it is not strange even if
someone gets the simple theory which explains mass”.
□- 48 -

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Fig.1

G(s)
Vin

Vout

β(s)

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Fig.2

X2

−( ict )2

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Fig. B=W

P

Vector P is generated by superpositionof three B=W circles.

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1/1 ページFig.3

z
P sinωjτ
P cosωjτ

P

ω jτ
y

x

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Fig.4

Ψ2

−( it )2

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Fig.7
Cл3

C(л2−л3)C(л3−л1)

V×Ж

v1×ξ*1

Cл2
C(л1−л2)Cл1

vj×ξ*j=C(лk−лn)file://C:\Documents%20and%20Settings\新美\デスクトップ\F.I.T\yoko.host\gaps-t.web\gaps-t.fig... 2005/12/13

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Fig.8

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I3

G3



R3

[2]
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I3

G3

R3

file://C:\Documents%20and%20Settings\新美\デスクトップ\F.I.T\yoko.host\gaps-t.web\gaps-t.fig... 2005/12/13

GAPS T. Sheet 1 (ωτ= Θe)wτ=ωτ/sinθθvalue of [ wτ][ real part ]R ωexp(±iρ)cosδ= [ imaginary part ]θ=asin(ωτ/wτ) R wsinθcos(wτcosθ)cos(wτsinθ)R ωexp(±iρ)cosδ= cωexp(+ρ)cosδ=i R wsinθsin(wτcosθ)cos(wτsinθ)ωexp(-ρ)cosδ=/ccwsinθexp(wτcosθ)cos(wτsinθ)

wsinθexp(-wτcosθ)cos(wτsinθ)00#DIV/0!

#DIV/0!

#DIV/0!

#DIV/0!

#DIV/0!

#DIV/0!

0.001

0.001

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.001001

0.001001

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.001005

0.001005

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.00101

0.00101

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.001020.00102

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.00105

0.00105

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.00110.0011

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.00120.0012

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.00150.0015

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.002

0.002

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.003

0.003

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.004

0.004

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.005

0.005

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.01

0.01

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.05

0.05

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

π/32

0.09817477

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

π/16

0.196349541

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

π/8

0.392699082

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

π/4

0.785398163

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

Θ

0.955316618

1.570796327

0.577350269

3.37867E-17

0.577350269

0.577350269

Θ'

0.955316618

1.570796285

0.577350269

2.32462E-08

0.577350292

0.577350246 0.999999919

0.96

1.471978367

0.571958749

0.054333298

0.63160841

0.522616408

0.97

1.396578918

0.560592186

0.095154844

0.672722699

0.480611169 0.714426865

0.98

1.345880466

0.549419179

0.122032563

0.700296097

0.452313503 0.645888938

0.99

1.305314832

0.538434371

0.143091124

0.722369803

0.429678317 0.594817661

111.270729093

0.527632619

0.16066663

0.741237521

0.410408187 0.553679725

π/3

1.047197551

1.148768441

0.478978107

0.219057481

0.808809151

0.342981044 0.424056829

1.07

1.103566123

0.456755146

0.238921505

0.834658096

0.318344422 0.381406978

1.1

1.052106942

0.42868808

0.260076352

0.865011729

0.290647133

1.12

1.021632472

0.41067342

0.271774998

0.883612936

0.274457624 0.310608427
0.252841727 0.277919978

B

π/2

#NUM!
1
0.82743738

0.33600369

1.15

0.980382018

0.384633258

0.286502499

0.909764489

1.170019129

0.955316618

0.36787825

0.294767392

0.926327125

0.239896054 0.258975526

1.27

0.851421497

0.290900989

0.3224704

1.002789845

0.188085814 0.187562544

1.351

0.785414239

0.235716287

0.333331452

1.061223242

0.157056516 0.147995737

1.37

0.771639616

0.223593362

0.334794073

1.074809295

0.150799834 0.140303805

1.47

0.70742029

0.164395223

0.337273614

1.146804193

0.122757905 0.107043474

1.52

0.679620352

0.1374606

0.335819001

1.183580736

0.11124701 0.093991906

1.570796327

0.653757546

0.11174974

0.332871812

1.221773346

0.100912046 0.082594735

1.6

0.639847215

0.097680322

0.330591236

1.244180345

0.095510278 0.076765622

1.67

0.608997992

0.065946825

0.323619909

1.29943283

0.083943415 0.064600042

1.77

0.570112671

0.025131944

0.310596354

1.38272882

0.070224695 0.050787034

1.87

0.536190121-0.010842846

0.294748388

1.471981573

0.059100047 0.040149991

1.97

0.506285747-0.042412154

0.276744742

1.568021604

0.049990665 0.031881363

2π/3

2.094395102

0.473641685-0.076069727

0.2521209

1.698161648

0.040839193 0.024049061

3π/42.35619449

0.417469923-0.128879952

0.195413014

2.017325515

0.027162839 0.013464777

2.38590.411954356-0.133470697

0.188748189

2.057930171

0.025967988 0.012618498

2.4

0.409388671-0.135556398

0.185576642

2.077545318

0.025421456 0.012236294

2.5

0.392096497-0.148677051

0.16299916

2.223239969

0.021893089

2.6

0.376243647-0.158999886

0.140429725

2.381197626

0.018898671 0.007936624
0.016348552 0.006404905

0.00984738

2.7

0.361653193-0.166702249

0.118068284

2.55250487

7π/8

2.748893572

0.354933152-0.169565878

0.107265607

2.641474223

0.01524092 0.005769854

15π/162.945243113

0.330334194-0.175553741

0.065195922

3.036773569

0.011548317 0.003802825

31π/32

3.043417883

0.319293633-0.175504095

0.045187092

3.259571628

0.010076036 0.003091215

π3.141592654

0.308979608-0.173625234

0.026022873

3.501081834

0.008803825 0.002514601

33π/32

3.239767424

0.299321533-0.170065299

0.007807352

3.762925654

0.007702294

17π/16

3.337942194

0.29025782-0.164971464-0.009367861

4.046868465

0.006746782 0.001667161

9π/8

3.534291735

0.2737039-0.150766461-0.040291189

4.688913737

0.005193933 0.001107705

3.6

0.268583229-0.144974936-0.049550558

4.928229168

0.004762966 0.000966466

3.7

0.261152019-0.13531384-0.062541908

5.318510742

0.00417811 0.000785579

3.8

0.25412552-0.124762882-0.074171384

5.742747004

0.003668483 0.000638803

5π/4

3.926990817

0.245735071-0.11031313-0.0869349

6.335236416

0.003113801 0.000491505

4π/3

4.188790205

0.230089887-0.077998663-0.106085326

7.775422869

0.002229832

4.20680.229087204-0.075697911-0.107049432

7.886659347

0.002179599 0.000276365

0.00204689

0.00028678

3π/2

4.71238898

0.204139377-0.011434267-0.116483173

11.81448683

0.001159515 9.81435E-05

5π/35.235987756

0.183479772

0.044458789-0.095496934

18.12924374

0.000612063 3.37611E-05

5.412810.177421162

0.05882824-0.083200706

20.9901511

0.000494666 2.35666E-05

7π/4

5.497787144

0.174650375

0.064764713-0.076616945

22.52863045

0.000446748 1.98302E-05

5.59

0.171740424

0.070461207-0.069069013

24.33137442

0.000400113 1.64443E-05

5.69

0.168692911

0.07574004-0.060494407

26.45650201

0.000355154 1.34241E-05

5.79

0.165752166

0.080068211-0.051608598

28.77464002

0.00031536 1.09596E-05

15π/8

5.890486225

0.162899082

0.083452559-0.042462648

31.31654202

0.000279961 8.93971E-06

31π/166.086835766

0.157599577

0.08728783-0.024325395

36.97543785

0.000222063

63π/326.185010537

0.15507758

0.087853732-0.015297634

40.19088881

0.000197863 4.92308E-06
0.000176351 4.03594E-06

6.0057E-06

2π6.283185307

0.152635344

0.087548167-0.006406743

43.69528591

2.05π

6.44026494

0.148884421

0.085325973

0.007341745

49.97031462

0.000146776 2.93726E-06

2.1π

6.597344573

0.145314075

0.081113154

0.020247931

57.17552186

0.000122243 2.13803E-06

2.15π

6.754424205

0.141911529

0.075105317

0.032050103

65.45145129

0.000101877 1.55653E-06

2.2π

6.911503838

0.138665188

0.067528748

0.042523387

74.96018288

8.49567E-05 1.13336E-06

2.25π

7.068583471

0.135564502

0.058634351

0.051482903

85.88866858

7.0888E-05 8.25348E-07

7.30130.131218223

0.043612115

0.061680836

105.157731

5.42665E-05 5.16049E-07

2.35π

7.382742736

0.129762463

0.037980578

0.064333574

112.9008143

4.94357E-05 4.37869E-07

2.4π

7.539822369

0.12704428

0.02678841

0.068070413

129.5205604

4.13154E-05 3.18988E-07

2.45π

7.696902001

0.124437931

0.015399927

0.069984674

148.643357

3.45459E-05 2.32408E-07

2.5π

7.853981634

0.121936635

0.004092991

0.070106443

170.6519487

2.8899E-05 1.69345E-07

2.55π

8.011061267

0.119534151-0.006867597

0.06850547

195.9882794

2.4186E-05 1.23405E-07

2.6π

8.168140899

0.117224724-0.017235414

0.065288152

225.1627324

2.02503E-05 8.99361E-08

2.65π8.325220532

0.115003039-0.026787173

0.060593832

258.7648218

1.6962E-05 6.55498E-08

8.52320.112320287-0.037363085

0.052835854

308.4878162

1.35747E-05

8.639379797

0.110803582-0.042688954

0.047470097

342.0818142

1.19146E-05 3.48296E-08
9.99127E-06 2.53912E-08

2.75π

4.4004E-08

2.8π8.79645943

0.108817027-0.04874138

0.039443272

393.4928192

2.85π

8.953539063

0.106900585-0.053386984

0.030734129

452.75944

8.3814E-06 1.85118E-08

2.9π9.110618695

0.105050601-0.056564692

0.021574678

521.0963633

7.03331E-06 1.34971E-08

2.95π

9.267698328

0.103263671-0.058249649

0.012199349

599.90784

5.90398E-06 9.84148E-09

3π9.424777961

0.10153662-0.058452584

0.002839608

690.81749

4.95756E-06 7.17636E-09
4.16412E-06 5.23326E-09

3.05π

9.581857593

0.099866482-0.05721842-0.006281191

795.7028214

3.1π

9.738937226

0.098250487-0.05462415-0.014952632

916.7352144

3.4987E-06 3.81648E-09

3.15π

9.896016859

0.096686037-0.050776082-0.022981623

1056.42624

2.94045E-06 2.78339E-09

3.2π

10.05309649

0.095170704-0.045806497-0.030196272

1217.681323

2.47195E-06 2.03005E-09

3.25π

10.21017612

0.093702205-0.039869825-0.036449178

1403.861914

2.07866E-06 1.48067E-09

10.42394

0.091775182-0.030549776-0.04320184

1704.365778

1.64266E-06 9.63794E-103.35π

10.52433539

0.090897275-0.025798143-0.045617803

1867.169263

1.47096E-06 7.87801E-103.4π10.68141502

0.089556938-0.018043525-0.048381537

2154.006498

1.23785E-06 5.74674E-10

3.45π

10.83849465

0.088255606-0.010073182-0.049881336

2485.399101

1.04193E-06 4.19221E-10

3.5π10.99557429

0.086991598-0.00208503-0.050117948

2868.327393

8.77221E-07

3.55π

11.15265392

0.08576333

0.005728269-0.049121935

3310.872852

7.38711E-07 2.23117E-10

3.6π

11.30973355

0.084569305

0.01318363-0.046952135

3822.392817

6.22205E-07 1.62779E-103.65π

11.46681319

0.083408108

0.020111541-0.043693525

4413.723055

5.24183E-07 1.18762E-10

11.65030.082091486

0.02733358-0.038654512

5222.417795

4.29168E-07 8.21781E-11
3.83305E-07 6.67259E-11

3.0583E-10

11.75410.081364928

0.030928415-0.035289067

5744.471506

3.75π

11.78097245

0.081178925

0.031795811-0.034363904

5887.996276

3.72258E-07 6.32231E-113.8π11.93805208

0.080108476

0.036310738-0.028567132

6802.310245

3.13798E-07 4.61311E-11

3.85π

12.09513172

0.079065918

0.039819866-0.022222649

7859.858653

2.64568E-07 3.36607E-113.9π

12.25221135

0.078050175

0.042264746-0.01549778

9083.238375

2.23102E-07

3.95π

12.40929098

0.077060226

0.043613746-0.008564593

10498.63191

1.88168E-07 1.79231E-11

2.4562E-114π12.56637061

0.076095097

0.04386212-0.00159575

12136.37915

1.58732E-07

4.05π

12.72345025

0.075153868

0.043031458

0.005239566

14031.64064

1.33923E-07 9.54436E-12

1.3079E-114.1π

12.88052988

0.074235659

0.041168533

0.011779541

16225.1672

1.1301E-07

4.15π

13.03760951

0.073339637

0.038343599

0.01787336

18764.1928

9.53779E-08 5.08297E-12

6.9651E-12

4.2π

13.19468915

0.072465005

0.034648171

0.0233845

21703.4707

8.05093E-08 3.70951E-12

4.25π

13.35176878

0.071611007

0.03019237

0.028193609

25106.47578

6.79689E-08 2.70722E-12

13.55537

0.070533611

0.02349245

0.033221779

30329.12287

5.45872E-08 1.79983E-124.35π

13.66592804

0.069962054

0.019514667

0.035328183

33609.77136

4.84652E-08

4.4π

13.82300768

0.069165753

0.013577351

0.037519969

38894.33445

4.09338E-08 1.05244E-12

1.442E-12

4.45π

13.98008731

0.068387389

0.007441626

0.038744527

45015.23128

3.45775E-08

4.5π

14.13716694

0.067626362

0.001260514

0.038993977

52105.53437

2.92122E-08 5.60636E-13

7.6813E-13

4.55π

14.29424657

0.066882099-0.004815281

0.038283976

60319.58666

2.46827E-08 4.09199E-134.6π

14.45132621

0.066154051-0.010640899

0.036652839

69836.41484

2.08582E-08 2.98672E-134.65π

14.60840584

0.065441695-0.016080633

0.034160138

80863.69218

1.76285E-08 2.18002E-13

14.78352

0.064665434-0.021539352

0.030462884

95233.74315

1.46159E-08 1.53474E-13

4.75π

14.9225651

0.064062067-0.025323217

0.026922985

108451.8693

1.25965E-08 1.16148E-13

4.8π

15.07964474

0.063393853-0.028925875

0.022385124

125616.5871

1.06499E-08 8.47807E-14

4.85π

15.23672437

0.062739445-0.031746363

0.017393306

145512.799

9.00511E-09 6.18853E-14

4.9π

15.393804

0.062098418-0.03373241

0.01207798

168577.1774

7.61522E-09 4.51735E-14
6.44057E-09 3.29751E-14

4.95π

15.55088364

0.061470366-0.034852875

0.006574708

195316.4792

5π15.70796327

0.060854899-0.035098068

0.001020828

226318.8325

5.4477E-09 2.40709E-14

5.05π

15.8650429

0.060251641-0.034479556-0.004447853

262266.847

4.60837E-09 1.75713E-14
3.89877E-09 1.28269E-14

5.1π

16.02212253

0.059660233-0.033029465-0.009700273

303952.8426

5.15π

16.17920217

0.05908033-0.030799316-0.01461315

352296.54

3.29876E-09 9.36359E-15

5.2π

16.3362818

0.058511597-0.02785842-0.019073779

408365.6119

2.79137E-09 6.83547E-15

5.25π

16.49336143

0.057953716-0.024291877-0.022982522

473399.5587

2.36226E-09 4.98999E-15

16.69062

0.057268034-0.019079504-0.026981238

570000.481

1.91581E-09 3.36107E-15

5.35π

16.8075207

0.056869289-0.015686934-0.028823551

636350.1479

1.69227E-09 2.65934E-15

5.4π

16.96460033

0.056342161-0.010875373-0.030639083

737877.7701

1.43252E-09 1.94141E-15

5.45π

17.12167996

0.05582472-0.005886064-0.031671365

855673.1932

1.21275E-09 1.41731E-15

5.5π

17.27875959

0.055316702-0.000843544-0.03190968

992352.6274

1.02679E-09

1.0347E-15

5.55π

17.43583923

0.054817851

0.004128629-0.03136267

1150954.381

8.6942E-10

7.5539E-165.6π

17.59291886

0.054327922

0.008910826-0.030057788

1335007.153

7.36231E-10 5.51481E-16

5.65π

17.74999849

0.053846676

0.013390147-0.0280403

1548609.412

6.23498E-10 4.02618E-16

17.9197

0.053336256

0.017769775-0.025131421

1818092.052

5.2107E-10 2.86603E-16

5.75π

18.06415776

0.05290933

0.021037463-0.022128875

2084273.557

4.47284E-10 2.14599E-16
3.78887E-10 1.56676E-16

5.8π

18.22123739

0.052452794

0.024035225-0.018400156

2418288.769

5.85π

18.37831702

0.052004074

0.026393292-0.014284828

2806029.628

3.20974E-10 1.14387E-16

5.9π

18.53539666

0.051562968

0.028065217-0.009889643

3256165.65

2.71934E-10 8.35135E-17

5.95π

18.69247629

0.051129287

0.029021934-0.005326313

3778769.3

2.30404E-10 6.09732E-17

6π18.84955592

0.050702842

0.029252174-0.000708738

4385543.592

1.95231E-10 4.45168E-17

6.05π

19.00663555

0.050283455

0.028762438

0.003849768

5090086.709

1.65439E-10 3.25022E-17

6.1π

19.16371519

0.049870952

0.027576545

0.008239197

5908199.655

1.40203E-10 2.37303E-17

6.15π

19.32079482

0.049465165

0.025734782

0.012355439

6858243.967

1.18826E-10

6.2π

19.47787445

0.049065931

0.023292659

0.016102695

7961557.633

1.00715E-10 1.26501E-17

6.25π

19.63495408

0.048673092

0.020319335

0.019395648

9242938.727

8.53697E-11

19.82787

0.048199157

0.016060841

0.022712026

11103108.7

6.9691E-11 6.27671E-18

1.7326E-17
9.2362E-18

6.35π

19.94911335

0.047905995

0.013112387

0.02434081

12459861.77

6.13498E-11 4.92379E-18

6.4π

20.10619298

0.047531446

0.009067144

0.025890141

14467834.61

5.20128E-11 3.59506E-18

6.45π

20.26327262

0.047162711

0.004862657

0.026781438

16800381.65

4.40996E-11 2.62492E-18

6.5π

20.42035225

0.046799655

0.000603846

0.02700318

19510108.24

3.73927E-11 1.91658E-18

6.55π

20.57743188

0.046442147-0.003604661

0.026560259

22658166.11

3.17078E-11

6.6π

20.73451151

0.046090062-0.007661059

0.025473611

26315644.78

2.68888E-11 1.02178E-18

6.65π

20.89159115

0.045743278-0.011468776

0.023779473

30565190.01

2.28036E-11 7.46064E-19

21.05754

0.045382539-0.015122898

0.021385783

35804410.45

1.91612E-11 5.35162E-19

1.3994E-18

6.75π

21.20575041

0.045065137-0.017991348

0.018783204

41240446.32

1.64037E-11 3.97758E-19

6.8π

21.36283004

0.044733554-0.020558395

0.01561851

47907758.8

1.39139E-11 2.90432E-19

6.85π

21.51990968

0.044406816-0.022584405

0.012117527

55655850.9

1.18027E-11 2.12066E-196.9π

21.67698931

0.044084818-0.024027864

0.008370538

64660335.65

6.95π

21.83406894

0.043767458-0.024862006

0.00447249

75125422.06

1.00124E-11 1.54846E-19
8.4941E-12 1.13066E-19

7π21.99114858

0.043454635-0.025075246

0.00052063

87288580.56

7.20643E-12 8.25587E-20

7.05π

22.14822821

0.043146254-0.024671247-0.003387875

101425970.9

6.11429E-12 6.02833E-20

7.1π22.30530784

0.042842221-0.023668613-0.007158293

117858757.7

5.18794E-12 4.40183E-20

7.15π

22.46238747

0.042542443-0.02210023-0.010700582

136960457.8

4.40217E-12 3.21419E-20

7.2π

22.61946711

0.042246833-0.020012272-0.013931501

159165490.7

3.73559E-12 2.34699E-20

7.25π

22.77654674

0.041955304-0.017462898-0.016776537

184979125.9

3.17011E-12 1.71377E-20

22.96632

0.041608422-0.013865917-0.019608425

221824065.6

2.60005E-12 1.17212E-20

7.35π

23.090706

0.041384155-0.011262791-0.021064341

249878884.7

2.28333E-12 9.13776E-21

7.4π

23.24778564

0.041104374-0.007773043-0.022415453

290443766.1

1.93797E-12 6.67246E-21

7.45π

23.40486527

0.040828352-0.004139743-0.023199252

337608682.7

1.64493E-127.5π

23.5619449

0.040556013-0.000453503-0.023404214

392449656.3

1.39626E-12 3.55781E-21

4.8723E-21

7.55π

23.71902453

0.040287285

0.003194983-0.023033046

456218457.2

1.18524E-12 2.59797E-21

7.6π

23.87610417

0.040022095

0.006717145-0.022102435

530371358.5

1.00616E-12 1.89708E-21

7.65π

24.0331838

0.039760374

0.010028649-0.020642456

616602600.2

8.54172E-13 1.38529E-21

24.19632

0.039492162

0.013161175-0.018611561

721061242.5

7.20614E-13

24.34734307

0.039247071

0.015715308-0.016315935

833506983.9

6.15688E-13 7.38672E-227.75π

9.9938E-22

7.8π

24.5044227

0.03899536

0.017959973-0.01356694

969141959.1

5.22754E-13 5.39398E-22

7.85π

24.66150233

0.038746857

0.019735937-0.010520552

1126893128

4.43866E-13 3.93884E-227.9π

24.81858196

0.038501502

0.021005794-0.007254958

1310373284

3.76899E-13 2.87627E-22

7.95π

24.9756616

0.038259235

0.021744926-0.00385269

1523786374

3.20048E-13 2.10035E-228π25.13274123

0.038020.021941949-0.00039857

1772024374

2.71784E-13 1.53375E-228.1π

25.44690049

0.037550395

0.020730629

0.006326995

2396678315

1.96017E-138.2π

25.76105976

0.037092252

0.017541568

0.012275782

3242004554

1.41394E-13 4.36133E-238.3π26.10557

0.036602532

0.012197659

0.017251072

4516111813

9.88422E-14 2.18866E-23

8.4π

26.38937829

0.036208712

0.006801468

0.019762912

5934809792

7.36052E-14 1.24023E-238.5π26.70353756

0.035782544

0.000353046

0.020651636

8031436594

5.31181E-14 6.61377E-24

8.6π

27.01769682

0.035366293-0.00597952

0.01951913

10870204665

3.83388E-14 3.52697E-24

27.33569

0.034954713-0.01164878

0.01647475

14768762068

2.75657E-14 1.86649E-248.8π27.64601535

0.03456219-0.015944503

0.011991495

19920288718

1.99808E-14 1.00304E-24

8.9π

27.96017462

0.034173699-0.018658869

0.006401223

26971589654

1.44274E-149π28.27433388

0.033793846-0.019504631

0.000314899

36523340945

1.04188E-14 2.85264E-25

9.1π

28.58849315

0.033422346-0.018441394-0.005668106

49463670139

7.52497E-15 1.52131E-25

9.2π28.90265241

0.033058926-0.015613761-0.010971568

66996628462

5.43557E-15

29.24528

0.03267148-0.010888213-0.015398987

93286012501

3.81281E-15 4.08723E-26

9.4π

29.53097094

0.032355296-0.006045381-0.017671635

1.22952E+11

2.83716E-15 2.30754E-26

9.5π

29.84513021

0.032014597-0.000282617-0.018478318

1.66589E+112.05012E-15 1.23065E-26

9.6π

30.15928947

0.031681

0.005387434-0.017476428

2.25739E+11

1.48158E-15 6.56325E-27

30.47561

0.03135206

0.010449319-0.014776838

3.06563E+11

1.06844E-15 3.48522E-27

9.8π30.78760801

0.031034239

0.014335643-0.010743724

4.1463E+11

7.74036E-16 1.86681E-27

9.9π

31.10176727

0.030720663

0.016783606-0.005726953

5.62023E+11

5.59563E-16 9.95623E-28

10π31.41592654

0.030413361

0.017554602-0.000255057

7.61887E+11

4.0456E-16 5.30997E-28≒10.3π32.38531

0.029502735

0.009833186

0.013905446

1.9493E+12

1.48798E-16 7.63341E-29
8.00396E-17 2.29147E-29

8.1787E-23

5.3491E-25

8.1132E-26

10.5π

32.98672286

0.02896469

0.00023134

0.016718833

3.49293E+12≒10.7π33.6158

0.028422505-0.009473164

0.013396502

6.43137E+12

4.18585E-17 6.50849E-30

11π34.55751919

0.02764777-0.015959022

0.000210784

1.6049E+13

1.58723E-17≒11.3π35.52566

0.026894132-0.008963919-0.012676283

4.1121E+13

5.86173E-18 1.42548E-31

9.8899E-31

11.5π

36.12831552

0.026445406-0.00019285-0.015265265

7.38892E+13

3.15426E-18

4.2689E-32≒11.7π36.75624

0.025993524

0.008663876-0.012251842

1.36112E+14

1.65431E-18

1.2154E-32

12π37.69911184

0.025343273

0.014629307-0.000177112

3.40811E+14

6.28054E-19 1.84282E-33≒12.3π38.66621

0.02470927

0.008235838

0.011646698

8.7428E+14

2.32734E-19

12.5π

39.26990817

0.024329338

0.000163225

0.014044216

1.57465E+15

1.25277E-19 7.95583E-35

2.662E-34≒12.7π39.8968

0.023946981-0.007981261

0.011287851

2.90162E+15

6.58653E-20 2.26995E-35

13π40.8407045

0.02339342-0.013504122

0.000150909

7.28674E+15

2.50296E-20 3.43495E-36≒13.3π41.80691

0.022852676-0.00761719-0.010771709

1.87115E+16

9.30185E-2113.5π

42.41150082

0.022526848-0.000139936-0.013004029

3.37684E+16

5.00836E-21 1.48315E-37≒13.7π4.9712E-37

43.0376

0.022199079

0.007399247-0.010463772

6.2248E+16

2.63847E-21 4.23864E-38

43.288296

0.022070496

0.009707743-0.008252412

7.95254E+16

2.04139E-21 2.56697E-38

14π43.98229715

0.021722189

0.01253965-0.000130118

1.56701E+17

1.00357E-21 6.40435E-39≒14.3π44.94779

0.021255518

0.007084409

0.010019356

4.0276E+17

3.73861E-22 9.28249E-40

14.5π

45.55309348

0.020973036

0.000121299

0.012107293

7.28073E+17

2.01355E-22 2.76559E-40≒14.7π46.17848

0.020688961-0.006896064

0.009752007

1.3425E+18

1.06262E-22 7.91527E-41

15π47.1238898

0.020273837-0.011703755

0.000113346

3.38676E+18

4.04488E-23 1.19432E-41≒15.3π48.08867

0.019867039-0.006622158-0.009364621

8.71097E+18

1.51015E-23 1.73362E-4215.5π

48.69468613

0.019619757-0.000106151-0.011326248

1.57713E+19

8.1347E-24/c ; Planck's c./light v.

48.886015

0.01954296

0.002042012-0.011096084

1.90228E+19

6.6916E-24 3.51767E-43

5.1579E-43≒15.7π49.31939

0.019371212

0.006456403-0.009131293

2.90864E+19

4.2998E-24 1.47829E-43

16π50.26548246

0.019006565

0.010972332-9.96196E-05

7.35183E+19

1.63771E-24 2.22763E-44≒16.3π51.22968

0.018648799

0.006216144

0.00879045

1.89219E+20

6.12585E-25 3.23745E-45

16.5π

51.83627878

0.018430542

9.36731E-05

0.010639864

3.43036E+20

3.3004E-25 9.62115E-46≒16.7π52.460448

0.018211233-0.006069994

0.00858444

6.3279E+20

1.74683E-25 2.76051E-46

17π53.40707511

0.017888408-0.01032695

8.82436E-05

1.60205E+21

6.65734E-26 4.15552E-47≒17.3π54.37081

0.017571299-0.005856592-0.00828291

4.1259E+21

2.49415E-26 6.04512E-48

17.5π

54.97787144

0.017377259-8.32728E-05-0.010031915

7.48825E+21

1.34406E-26 1.79489E-48≒17.7π55.60159

0.017182308

0.005727408-0.008099239

1.38165E+22

7.12197E-27 5.15468E-49

18π

56.54866776

0.016894511

0.009753269-7.87105E-05

3.50295E+22

2.71578E-27 7.75284E-50≒18.3π57.51189

0.016611531

0.005537151

0.007830229

9.02567E+22

1.01901E-27 1.12901E-50

18.5π

58.11946409

0.01643786

7.45132E-05

0.009489683

1.6399E+23

5.49177E-28 3.34884E-51≒18.7π58.74269

0.016263449-0.005420739

0.007666446

3.02611E+23

2.91327E-28

19π59.69026042

0.016005248-0.00923997

7.06428E-05

7.6827E+23

1.11135E-28 1.44657E-52≒19.3π60.65307

0.015751159-0.005250398-0.007424698

1.98043E+24

4.17549E-29 2.10838E-53

19.5π

61.26105675

0.015594823-6.70664E-05-0.009003061

3.60169E+24

2.2506E-29 6.24874E-54≒19.7π61.88388

0.015437858

0.005145525-0.007277345

6.64713E+24

1.19504E-29 1.79784E-54

20π62.83185307

0.015204922

0.008777996-6.37548E-05

1.68958E+25

4.56073E-30 2.69932E-55

628.3185307

0.001520434

0.000877823-6.37517E-07

6.5821E+269

1.1707E-276

3.02169E+25

3.16153E-26

0.961835πE+25

#NUM!

#NUM!

#NUM!

Graph.1

Graph.2

Graph.30Graph.4

9.6271E-52

0
#NUM!

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2005/12/13

case ①(wτ=48.87668)

ωτ= wτsinθ

value of [ωτ]0[c]

θ=asin(ωτ/wτ)ωexp(+ρ)cosδ=[

wsinθ exp(wτcosθ)cos(wτsinθ)]ωexp(-ρ)cosδ=

ratio/c=3.51767E-43

wsinθexp(-wτcosθ)cos(wτsinθ)

0

0

0

0

π/2×0.00000000000553261063

8.6906E-12

1.77807E-13

299792458.2

1.05457E-34

3.51767E-43

π/2×0.00000002927 57

4.59862E-08

9.40861E-10

1.58635E+12

5.58024E-31

3.51767E-43

π/2×0.00001312066853#DIV/0!

2.06099E-05

4.21671E-07

7.10962E+14

2.50093E-28

3.51767E-43

0.001

0.001

2.04597E-05

3.44961E+16

1.21346E-26

3.51767E-43

0.001001

0.001001

2.04801E-05

3.45306E+16

1.21467E-26

3.51767E-43

0.001005

0.001005

2.0562E-05

3.46686E+16

1.21953E-26

3.51767E-43

0.00101

0.00101

2.06643E-05

3.48411E+16

1.2256E-26

3.51767E-43

0.001020.001022.08688E-05

3.51861E+16

1.23773E-26

3.51767E-43

0.00105

0.00105

2.14826E-05

3.62209E+16

1.27413E-26

3.51767E-43

0.00110.00112.25056E-05

3.79457E+16

1.33481E-26

3.51767E-43

0.00120.00122.45516E-05

4.13954E+16

1.45615E-26

3.51767E-43

0.00150.00153.06895E-05

5.17442E+16

1.82019E-26

3.51767E-43
3.51767E-43

0.002

0.002

4.09193E-05

6.89922E+16

2.42692E-26

0.003

0.003

6.1379E-05

1.03488E+17

3.64037E-26

3.51767E-43

0.004

0.004

8.18386E-05

1.37984E+17

4.85381E-26

3.51767E-43

0.005

0.005

0.000102298

1.72479E+17

6.06723E-26

3.51767E-43

0.01

0.01

0.000204597

3.44944E+17

1.2134E-25

3.51768E-43

0.05
π/32

π/16

π/8

π/4

0.05

0.001022983

1.72261E+18

6.05988E-25

3.51785E-43

0.09817477

0.002008623

3.37001E+18

1.18569E-24

3.51836E-43

0.12

0.002455161

4.10916E+18

1.44589E-24

3.51871E-43

0.15

0.003068953

5.11514E+18

1.80017E-24

3.51929E-43

0.17

0.003478148

5.7781E+18

2.03375E-24

3.51975E-43

0.196349541

0.004017255

6.64054E+18

2.33777E-24

3.52045E-43

0.25

0.005114936

8.3506E+18

2.94122E-24

3.52217E-43

0.3

0.006137935

9.87753E+18

3.48099E-24

3.52415E-43

0.35

0.00716094

1.13275E+19

3.99463E-24

3.5265E-43

0.392699082

0.008034574

1.24957E+19

4.40947E-24

3.52879E-43

0.47

0.009616186

1.44225E+19

5.09636E-24

3.53361E-43

0.55

0.011253048

1.61249E+19

5.70741E-24

3.53951E-43

0.63

0.012889939

1.74894E+19

6.20235E-24

3.54635E-43

0.71

0.014526866

1.84785E+19

6.56751E-24

3.55414E-43

0.785398163

0.016069667

1.90373E+19

6.78175E-24

3.56235E-43

0.81

0.016573079

1.9137E+19

6.82275E-24

3.56521E-43

0.84

0.017186956

1.92018E+19

6.85279E-24

3.56882E-43

0.87

0.01780084

1.9203E+19

6.86043E-24

3.57257E-43

0.90.01841473

1.91396E+19

6.84519E-24

3.57646E-43

0.93

0.019028627

1.90104E+19

6.80663E-24

3.58048E-43

0.935

0.019130944

1.89824E+19

6.79791E-24

3.58116E-43

0.94

0.019233261

1.89526E+19

6.78853E-24

3.58185E-43

0.945

0.019335579

1.89209E+19

6.77848E-24

3.58254E-43

0.95

0.019437896

1.88873E+19

6.76777E-24

3.58323E-43

0.955

0.019540214

1.88519E+19

6.7564E-24

3.58393E-43

0.95510.01954226

1.88512E+19

6.75617E-24

3.58395E-43

0.95520.019544306

1.88505E+19

6.75593E-24

3.58396E-43

Θe0.955316618

0.019546693

1.88496E+19

6.75566E-24

3.58398E-43

Θe×1.00000000000001

0.955316618

0.019546693

1.88496E+19

6.75566E-24

3.58398E-43

0.96

0.019642532

1.88146E+19

6.74436E-24

3.58463E-43

1
π/3

B

π/2×0.999999999890292

0.97

0.019847168

1.87344E+19

6.71826E-24

3.58605E-43

0.98

0.020051805

1.86467E+19

6.68947E-24

3.58748E-43

0.99

0.020256443

1.85515E+19

6.65798E-24

3.58893E-43

10.020461082

1.84486E+19

6.62378E-24

3.59039E-43

1.047197551

0.02142694

1.78606E+19

6.42535E-24

3.5975E-43

1.07

0.02189358

1.75154E+19

6.3074E-24

3.60105E-43

1.1

0.022507521

1.70003E+19

6.13005E-24

3.60585E-43

1.12

0.022916819

1.66182E+19

5.99774E-24

3.60913E-43
3.61416E-43

1.15

0.023530774

1.59871E+19

5.77802E-24

1.170019129

0.023940474

1.55273E+19

5.61714E-24

3.6176E-43

1.27

0.025986686

1.27677E+19

4.64195E-24

3.63571E-43

1.351

0.027644515

9.97318E+18

3.64175E-24

3.65154E-43

1.37

0.028033399

9.24665E+18

3.38003E-24

3.65541E-43

1.47

0.030080229

4.99108E+18

1.83508E-24

3.67672E-43

1.48

0.030284918

4.52658E+18

1.6653E-24

3.67894E-43

1.49

0.030489609

4.05518E+18

1.49278E-24

3.68118E-43

1.50.030694302

3.57694E+18

1.31754E-24

3.68343E-43

1.51

0.030898995

3.0919E+18

1.13958E-24

3.6857E-43

1.52

0.03110369

2.60012E+18

9.58919E-25

3.68799E-43

1.53

0.031308386

2.10165E+18

7.75569E-25

3.69029E-43

1.54

0.031513084

1.59655E+18

5.89544E-25

3.69261E-43
3.69495E-43

1.55

0.031717783

1.08488E+18

4.00857E-25

1.56

0.031922483

5.66694E+17

2.09523E-25

3.6973E-43

1.570796327

0.032143485

9105079932

3.36875E-33

3.69986E-43

π/2×0.999999999996476

1.570796327

0.032143485

292477868.7

1.08213E-34

3.69986E-43

π/2

1.570796327

0.032143485

3236.565649

1.19748E-39

3.69986E-43

case ②(wτ=48.886015)

ωτ= wτsinθ

value of [ωτ]0[c]

θ=asin(ωτ/wτ)ωexp(+ρ)cosδ=[ ]wsinθ exp(wτcosθ)cos(wτsinθ)ωexp(-ρ)cosδ=

ratio/c=3.51767E-43

wsinθexp(-wτcosθ)cos(wτsinθ)

0

0

0

0

π/2×0.00000000000553261063

8.6906E-12

1.77773E-13

302546340.3

1.04457E-34

3.4526E-43

π/2×0.00000002927 57

4.59862E-08

9.40681E-10

1.60092E+12

5.52734E-31

3.4526E-43

π/2×0.00001312066853

2.06099E-05

4.21591E-07

7.17493E+14

2.47722E-28

3.4526E-43

0.001

2.04557E-05

3.4813E+16

1.20196E-26

3.45261E-43

0.001#DIV/0!

0.001001

0.001001

2.04762E-05

3.48478E+16

1.20316E-26

3.45261E-43

0.001005

0.001005

2.0558E-05

3.49871E+16

1.20797E-26

3.45261E-43

0.00101

0.00101

2.06603E-05

3.51611E+16

1.21398E-26

3.45261E-43

0.001020.001022.08649E-05

3.55093E+16

1.226E-26

3.45261E-43

0.00105

0.00105

2.14785E-05

3.65537E+16

1.26205E-26

3.45261E-43

0.00110.00112.25013E-05

3.82943E+16

1.32215E-26

3.45261E-43

0.00120.00122.45469E-05

4.17756E+16

1.44235E-26

3.45261E-43

0.00150.00153.06836E-05

5.22195E+16

1.80293E-26

3.45261E-43

0.002

0.002

4.09115E-05

6.96259E+16

2.40391E-26

3.45261E-43

0.003

0.003

6.13672E-05

1.04439E+17

3.60585E-26

3.45261E-43

0.004

0.004

8.1823E-05

1.39251E+17

4.80779E-26

3.45261E-43

0.005

0.005

0.000102279

1.74063E+17

6.00971E-26

3.45261E-43

0.01

0.01

0.000204557

3.48113E+17

1.2019E-25

3.45261E-43
3.45278E-43

0.05
π/32

π/16

π/8

π/4

0.05

0.001022788

1.73843E+18

6.00243E-25

0.09817477

0.00200824

3.40097E+18

1.17445E-24

3.45329E-43

0.12

0.002454692

4.14691E+18

1.43219E-24

3.45362E-43

0.15

0.003068367

5.16213E+18

1.7831E-24

3.45419E-43

0.17

0.003477484

5.83118E+18

2.01447E-24

3.45465E-43

0.196349541

0.004016488

6.70154E+18

2.3156E-24

3.45533E-43

0.25

0.005113959

8.42731E+18

2.91334E-24

3.45702E-43

0.3

0.006136763

9.96827E+18

3.44799E-24

3.45897E-43

0.35

0.007159573

1.14315E+19

3.95675E-24

3.46127E-43

0.392699082

0.00803304

1.26105E+19

4.36766E-24

3.46351E-43

0.47

0.00961435

1.4555E+19

5.04804E-24

3.46824E-43

0.55

0.011250899

1.6273E+19

5.6533E-24

3.47404E-43

0.63

0.012887478

1.765E+19

6.14354E-24

3.48075E-43
3.48839E-43

0.71

0.014524092

1.86482E+19

6.50523E-24

0.785398163

0.016066598

1.92122E+19

6.71744E-24

3.49645E-43

0.81

0.016569914

1.93128E+19

6.75805E-24

3.49926E-43

0.84

0.017183674

1.93782E+19

6.78781E-24

3.5028E-43

0.87

0.01779744

1.93795E+19

6.79537E-24

3.50648E-43

0.90.018411213

1.93154E+19

6.78028E-24

3.51029E-43

0.93

0.019024993

1.91851E+19

6.74209E-24

3.51424E-43

0.935

0.019127291

1.91568E+19

6.73345E-24

3.51491E-43

0.94

0.019229588

1.91267E+19

6.72416E-24

3.51558E-43

0.945

0.019331886

1.90947E+19

6.71421E-24

3.51626E-43

0.95

0.019434184

1.90609E+19

6.7036E-24

3.51694E-43

0.955

0.019536482

1.90251E+19

6.69233E-24

3.51763E-43

0.95510.019538528

1.90244E+19

6.6921E-24

3.51764E-43

0.95520.019540574

1.90237E+19

6.69187E-24

3.51766E-43

Θe0.955316618

0.01954296

1.90228E+19

6.6916E-24

3.51767E-43

Θe×1.00000000000001

0.955316618

0.01954296

1.90228E+19

6.6916E-24

3.51767E-43

0.96

0.01963878

1.89875E+19

6.6804E-24

3.51832E-43

0.97

0.019843378

1.89066E+19

6.65455E-24

3.51971E-43

0.98

0.020047976

1.8818E+19

6.62604E-24

3.52111E-43

1
π/3

B

0.99

0.020252575

1.87219E+19

6.59485E-24

3.52253E-43

10.020457175

1.86181E+19

6.56097E-24

3.52397E-43

1.047197551

0.021422848

1.80247E+19

6.36442E-24

3.53094E-43

1.07

0.021889398

1.76764E+19

6.24759E-24

3.53443E-43

1.1

0.022503222

1.71565E+19

6.07192E-24

3.53914E-43

1.12

0.022912442

1.67709E+19

5.94086E-24

3.54236E-43

1.15

0.02352628

1.6134E+19

5.72322E-24

3.5473E-43

1.170019129

0.023935902

1.56699E+19

5.56387E-24

3.55067E-43
3.56844E-43

1.27

0.025981723

1.2885E+19

4.59792E-24

1.351

0.027639234

1.00648E+19

3.60721E-24

3.58397E-43

1.37

0.028028044

9.33162E+18

3.34797E-24

3.58777E-43

1.47

0.030074483

5.03695E+18

1.81767E-24

3.60868E-43

1.48

0.030279134

4.56818E+18

1.64951E-24

3.61086E-43

1.49

0.030483785

4.09245E+18

1.47862E-24

3.61305E-43

1.50.030688439

3.60981E+18

1.30504E-24

3.61527E-43

1.51

0.030893093

3.12031E+18

1.12877E-24

3.61749E-43
3.61974E-43

1.52

0.031097749

2.62401E+18

9.49824E-25

1.53

0.031302406

2.12096E+18

7.68213E-25

3.622E-43

1.54

0.031507064

1.61122E+18

5.83952E-25

3.62427E-43

1.55

0.031711724

1.09485E+18

3.97055E-25

3.62657E-43

1.56

0.031916385

5.71902E+17

2.07536E-25

3.62888E-43

π/2×0.999999999890292

1.570796327

0.032137345

9188775010

3.3368E-33

3.63139E-43

π/2×0.999999999996476

1.570796327

0.032137345

295142303.7

1.07178E-34

3.63139E-43

π/2

1.570796327

0.032137345

15105.97209

5.48556E-39

3.63139E-43

case ③(wτ=48.901915)

ωτ= wτsinθ

value of [ωτ]0[c]

θ=asin(ωτ/wτ)ωexp(+ρ)cosδ=[ ]wsinθ exp(wτcosθ)cos(wτsinθ)ωexp(-ρ)cosδ=

ratio/c=3.51767E-43

wsinθexp(-wτcosθ)cos(wτsinθ)

0

0

0

0

π/2×0.00000000000553261063

8.6906E-12

1.77715E-13

307295327.3

1.02776E-34

3.34454E-43

π/2×0.00000002927 57

4.59862E-08

9.40375E-10

1.62605E+12

5.43838E-31

3.34454E-43

π/2×0.00001312066853

2.06099E-05

4.21454E-07

7.28755E+14

2.43735E-28

3.34454E-43

0.001

0.001

2.04491E-05

3.53595E+16

1.18261E-26

3.34454E-43

0.001001

0.001001

2.04695E-05

3.53948E+16

1.18379E-26

3.34454E-43

0.001005

0.001005

2.05513E-05

3.55363E+16

1.18852E-26

3.34454E-43

0.00101

0.00101

2.06536E-05

3.57131E+16

1.19444E-26

3.34454E-43

0.001020.001022.08581E-05

3.60667E+16

1.20626E-26

3.34454E-43

0.00105

0.00105

2.14716E-05

3.71274E+16

1.24174E-26

3.34454E-43

0.00110.00112.2494E-05

3.88954E+16

1.30087E-26

3.34454E-43

0.00120.00122.45389E-05

4.24314E+16

1.41913E-26

3.34454E-43

0.00150.0015

3.06736E-05

5.30392E+16

1.77392E-26

3.34454E-43

0.002

0.002

4.08982E-05

7.07188E+16

2.36522E-26

3.34454E-43

0.003

0.003

6.13473E-05

1.06078E+17

3.54782E-26

3.34454E-43

0.004

0.004

8.17964E-05

1.41437E+17

4.73041E-26

3.34454E-43

0.005

0.005

0.000102245

1.76795E+17

5.91299E-26

3.34454E-43

0.01

0.01

0.000204491

3.53577E+17

1.18255E-25

3.34455E-43

0.05

0.05

0.001022455

1.76572E+18

5.90582E-25

3.34471E-43

0.09817477

0.002007587

3.45435E+18

1.15555E-24

3.3452E-43

0.120.002453894

4.212E+18

1.40914E-24

3.34552E-43

0.15

0.003067369

5.24316E+18

1.7544E-24

3.34608E-43

0.17

0.003476353

5.92271E+18

1.98205E-24

3.34652E-43

0.196349541

0.004015182

6.80673E+18

2.27833E-24

3.34718E-43

0.25

0.005112297

8.55959E+18

2.86645E-24

3.34882E-43

0.3

0.006134768

1.01247E+19

3.3925E-24

3.3507E-43

0.35

0.007157245

1.1611E+19

3.89307E-24

3.35293E-43

0.392699082

0.008030428

1.28084E+19

4.29737E-24

3.3551E-43

0.47

0.009611224

1.47835E+19

4.96679E-24

3.35968E-43

0.55

0.01124724

1.65285E+19

5.56231E-24

3.36529E-43

0.63

0.012883287

1.79271E+19

6.04465E-24

3.3718E-43

0.71

0.014519369

1.8941E+19

6.40053E-24

3.3792E-43

0.785398163

0.016061374

1.95138E+19

6.60932E-24

3.387E-43

0.81

0.016564526

1.9616E+19

6.64927E-24

3.38972E-43

0.840.017178086

1.96824E+19

6.67855E-24

3.39315E-43

0.87

0.017791653

1.96837E+19

6.68599E-24

3.39671E-43

0.90.018405226

1.96187E+19

6.67114E-24

3.4004E-43

0.93

0.019018807

1.94863E+19

6.63356E-24

3.40422E-43

0.935

0.019121071

1.94576E+19

6.62506E-24

3.40487E-43

0.94

0.019223335

1.9427E+19

6.61592E-24

3.40553E-43

0.945

0.0193256

1.93945E+19

6.60613E-24

3.40618E-43

0.95

0.019427864

1.93601E+19

6.59569E-24

3.40684E-43

0.955

0.019530129

1.93238E+19

6.5846E-24

3.40751E-43

0.95510.019532174

1.93231E+19

6.58438E-24

3.40752E-43

0.95520.01953422

1.93223E+19

6.58415E-24

3.40753E-43

Θe0.955316618

0.019536605

1.93215E+19

6.58388E-24

3.40755E-43

Θe×1.00000000000001

0.955316618

0.019536605

1.93215E+19

6.58388E-24

3.40755E-43

0.96

0.019632394

1.92856E+19

6.57287E-24

3.40817E-43

0.97

0.019836925

1.92034E+19

6.54743E-24

3.40952E-43

0.98

0.020041457

1.91135E+19

6.51938E-24

3.41088E-43

0.99

0.020245989

1.90158E+19

6.48869E-24

3.41225E-43

10.020450522

1.89105E+19

6.45535E-24

3.41364E-43

1.047197551

0.021415881

1.83077E+19

6.26197E-24

3.4204E-43

1.07

0.02188228

1.79539E+19

6.14701E-24

3.42378E-43

π/32

π/16

π/8

π/4

1
π/3

B

#DIV/0!

1.1

0.022495904

1.74259E+19

5.97418E-24

3.42834E-43

1.12

0.022904991

1.70343E+19

5.84522E-24

3.43145E-43

1.15

0.02351863

1.63874E+19

5.63109E-24

3.43624E-43

1.170019129

0.023928118

1.5916E+19

5.4743E-24

3.4395E-43

1.27

0.025973273

1.30873E+19

4.5239E-24

3.45671E-43

1.351

0.027630245

1.02229E+19

3.54913E-24

3.47175E-43

1.37

0.028018929

9.47816E+18

3.29407E-24

3.47543E-43

1.47

0.030064701

5.11605E+18

1.78841E-24

3.49568E-43

1.48

0.030269286

4.63992E+18

1.62295E-24

3.49779E-43

1.49

0.030473871

4.15672E+18

1.45482E-24

3.49991E-43

1.50.030678457

3.6665E+18

1.28403E-24

3.50206E-43

1.51

0.030883045

3.16932E+18

1.1106E-24

3.50421E-43

1.52

0.031087634

2.66522E+18

9.3453E-25

3.50639E-43

1.53

0.031292225

2.15427E+18

7.55843E-25

3.50858E-43

1.54

0.031496817

1.63653E+18

5.74549E-25

3.51078E-43

1.55

0.03170141

1.11204E+18

3.90661E-25

3.513E-43

1.56

0.031906004

5.80884E+17

2.04194E-25

3.51524E-43

π/2×0.999999999890292

1.570796327

0.032126893

9333097359

3.28307E-33

3.51767E-43

π/2×0.999999999996476

1.570796327

0.032126893

299801584.4

1.0546E-34

3.51767E-43

π/2

1.570796327

0.032126893

15343.21238

5.39723E-39

3.51767E-43

case ④(wτ=48.87668)

ωτ= wτsinθ

value of [ωτ]0[c]

ω2exp(+ρ)cosδ/(σ 2+ω2)=[ ]sin2θexp(wτcosθ)cos(wτsinθ)θ=asin(ωτ/wτ)ω2exp(-ρ)cosδ/(σ 2+ω2)=

ratio/c=3.51767E-43

sin2θexp(-wτcosθ)cos(wτsinθ)

0

0

0

0

π/2×0.00000000000553261063

8.6906E-12

1.77807E-13

5.33051E-05

1.8751E-47

3.51767E-43

π/2×0.00000002927 57

4.59862E-08

9.40861E-10

1492.53097

5.25023E-40

3.51767E-43

π/2×0.00001312066853

2.06099E-05

4.21671E-07

299792458

1.05457E-34

3.51767E-43

0.001

0.001

2.04597E-05

7.05779E+112.4827E-31

3.51767E-43

0.001001

0.001001

2.04801E-05

7.07191E+11

2.48767E-31

3.51767E-43

0.001005

0.001005

2.0562E-05

7.12855E+11

2.50759E-31

3.51767E-43

0.00101

0.00101

2.06643E-05

7.19965E+11

2.5326E-31

3.51767E-43

0.001020.001022.08688E-05

7.34293E+11

2.583E-31

3.51767E-43

0.00105

0.00105

2.14826E-05

7.78121E+11

2.73717E-31

3.51767E-43

0.00110.00112.25056E-05

8.53993E+11

3.00406E-31

3.51767E-43

0.00120.00122.45516E-05

1.01632E+12

3.57508E-31

3.51767E-43

0.00150.00153.06895E-05

1.588E+12

5.58607E-31

3.51767E-43

0.002

0.002

4.09193E-05

2.82311E+12

9.93078E-31

3.51767E-43

0.003

0.003

6.1379E-05

6.35199E+122.23442E-30

3.51767E-43

0.004

0.004

8.18386E-05

1.12924E+13

3.97229E-30

3.51767E-43

0.005

0.005

0.000102298

1.76443E+13

6.20667E-30

3.51767E-43

0.01

0.01

0.000204597

7.05743E+13

2.48258E-29

3.51768E-43

0.05

0.05

0.001022983

1.7622E+15

6.19915E-28

3.51785E-43

0.09817477

0.002008623

6.76908E+15

2.38161E-27

3.51836E-43

0.12

0.002455161

1.00887E+16

3.5499E-27

3.51871E-43

0.15

0.003068953

1.56981E+16

5.52462E-27

3.51929E-43

0.17

0.003478148

2.00971E+16

7.07366E-27

3.51975E-43

0.196349541

0.004017255

2.66767E+16

9.39138E-27

3.52045E-43

0.25

0.005114936

4.27126E+16

1.50441E-26

3.52217E-43

0.3

0.006137935

6.06273E+16

2.1366E-26

3.52415E-43

0.35

0.00716094

8.11146E+16

2.8605E-26

3.5265E-43

0.392699082

0.008034574

1.00397E+17

3.54278E-26

3.52879E-43

0.47

0.009616186

1.38688E+17

4.90068E-26

3.53361E-43

0.55

0.011253048

1.8145E+17

6.42245E-26

3.53951E-43

0.63

0.012889939

2.25431E+17

7.99456E-26

3.54635E-43

0.71

0.014526866

2.68425E+17

9.54019E-26

3.55414E-43

0.785398163

0.016069667

3.0591E+17

1.08976E-25

3.56235E-43

0.81

0.016573079

3.17145E+17

1.13069E-25

3.56521E-43

0.84

0.017186956

3.30004E+17

1.17773E-25

3.56882E-43

0.87

0.01780084

3.41812E+17

1.22115E-25

3.57257E-43

0.90.01841473

3.5243E+17

1.26045E-25

3.57646E-43

0.93

0.019028627

3.6172E+17

1.29513E-25

3.58048E-43

0.935

0.019130944

3.6313E+17

1.30043E-25

3.58116E-43

0.94

0.019233261

3.64498E+17

1.30558E-25

3.58185E-43

0.945

0.019335579

3.65824E+17

1.31058E-25

3.58254E-43

0.95

0.019437896

3.67107E+17

1.31543E-25

3.58323E-43

0.955

0.019540214

3.68347E+17

1.32013E-25

3.58393E-43

0.95510.01954226

3.68371E+17

1.32022E-25

3.58395E-43

0.95520.019544306

3.68396E+17

1.32032E-25

3.58396E-43

Θe0.955316618

0.019546693

3.68424E+17

1.32042E-25

3.58398E-43

Θe×1.00000000000001

0.955316618

0.019546693

3.68424E+17

1.32042E-25

3.58398E-43

0.96

0.019642532

3.69543E+17

1.32468E-25

3.58463E-43

0.97

0.019847168

3.71801E+17

1.3333E-25

3.58605E-43

0.98

0.020051805

3.73875E+17

1.34127E-25

3.58748E-43

0.99

0.020256443

3.75761E+17

1.34858E-25

3.58893E-43

10.020461082

3.77453E+17

1.3552E-25

3.59039E-43

1.047197551

0.02142694

3.82669E+17

1.37665E-25

3.5975E-43

1.07

0.02189358

3.83445E+17

1.38081E-25

3.60105E-43

π/32

π/16

π/8

π/4

1
π/3

#DIV/0!

1.1

0.022507521

3.82602E+17

1.37961E-25

3.60585E-43

1.12

0.022916819

3.80804E+17

1.37437E-25

3.60913E-43

1.15

0.023530774

3.76155E+17

1.35949E-25

3.61416E-43

1.170019129

0.023940474

3.71694E+17

1.34464E-25

3.6176E-43

1.27

0.025986686

3.31752E+17

1.20615E-25

3.63571E-43

1.351

0.027644515

2.75669E+17

1.00661E-25

3.65154E-43

1.37

0.028033399

2.59181E+17

9.47412E-26

3.65541E-43

1.47

0.030080229

1.5011E+17

5.51913E-26

3.67672E-43

1.48

0.030284918

1.37066E+17

5.04258E-26

3.67894E-43

1.49

0.030489609

1.23622E+17

4.55073E-26

3.68118E-43

1.50.030694302

1.09774E+17

4.04346E-26

3.68343E-43

1.51

0.030898995

9.55213E+16

3.52063E-26

3.6857E-43

1.52

0.03110369

8.08602E+16

2.98211E-26

3.68799E-43

1.53

0.031308386

6.57885E+16

2.42779E-26

3.69029E-43

1.54

0.031513084

5.03039E+16

1.85753E-26

3.69261E-43

1.55

0.031717783

3.44042E+16

1.27122E-26

3.69495E-43

1.56

0.031922483

1.80872E+16

6.68737E-27

3.6973E-43

π/2×0.999999999890292

1.570796327

0.032143485

292618608.9

1.08265E-34

3.69986E-43

π/2×0.999999999996476

1.570796327

0.032143485

9399639.294

3.47773E-36

3.69986E-43

π/2

1.570796327

0.032143485

104.0165869

3.84847E-41

3.69986E-43

Bcase ⑤(wτ=48.886015)

ωτ= wτsinθ

value of [ωτ]0[c]

ω2exp(+ρ)cosδ/(σ 2+ω2)=[ ]sin2θexp(wτcosθ)cos(wτsinθ)θ=asin(ωτ/wτ)ω2exp(-ρ)cosδ/(σ 2+ω2)=

ratio/c=3.51767E-43

sin2θexp(-wτcosθ)cos(wτsinθ)

0

0

0

0

π/2×0.00000000000553261063

8.6906E-12

1.77773E-13

5.37845E-05

1.85697E-47

π/2×0.00001312066853

2.06099E-05

4.21591E-07

302488567.5

1.04437E-34

3.4526E-43

0.001

0.001

2.04557E-05

7.12126E+11

2.45869E-31

3.45261E-43

0.001001

0.001001

2.04762E-05

7.13551E+11

2.46361E-31

3.45261E-43

0.001005

0.001005

2.0558E-05

7.19265E+11

2.48334E-31

3.45261E-43

0.00101

0.00101

2.06603E-05

7.2644E+11

2.50811E-31

3.45261E-43

0.001020.001022.08649E-05

7.40896E+11

2.55802E-31

3.45261E-43

0.00105

0.00105

2.14785E-05

7.85119E+112.71071E-31

3.45261E-43

0.00110.00112.25013E-05

8.61673E+11

2.97502E-31

3.45261E-43

0.00120.00122.45469E-05

1.02546E+12

3.54051E-31

3.45261E-43

0.00150.00153.06836E-05

1.60228E+12

5.53205E-31

3.45261E-43

0.002

0.002

4.09115E-05

2.8485E+12

9.83475E-31

3.45261E-43

0.003

0.003

6.13672E-05

6.40911E+12

2.21281E-30

3.45261E-43

0.004

0.004

8.1823E-05

1.13939E+13

3.93388E-30

3.45261E-43

0.005

0.005

0.000102279

1.78029E+13

6.14666E-30

3.45261E-43

0.01

0.01

0.000204557

7.1209E+13

2.45857E-29

3.45261E-43

0.05

0.05

0.001022788

1.77805E+15

6.13921E-28

3.45278E-43

0.09817477

0.00200824

6.82996E+15

2.35858E-27

3.45329E-43

0.12

0.002454692

1.01794E+16

3.51557E-27

3.45362E-43

0.15

0.003068367

1.58393E+16

5.4712E-27

3.45419E-43

0.17

0.003477484

2.02778E+16

7.00526E-27

3.45465E-43

0.196349541

0.004016488

2.69166E+16

9.30056E-27

3.45533E-43

π/32

π/16

π/8

π/4

#DIV/0!
3.4526E-43

0.25

0.005113959

4.30967E+16

1.48986E-26

3.45702E-43

0.3

0.006136763

6.11725E+16

2.11594E-26

3.45897E-43

0.35

0.007159573

8.18441E+16

2.83284E-26

3.46127E-43

0.392699082

0.00803304

1.013E+17

3.50852E-26

3.46351E-43

0.47

0.00961435

1.39935E+17

4.85328E-26

3.46824E-43

0.55

0.011250899

1.83082E+17

6.36034E-26

3.47404E-43

0.63

0.012887478

2.27458E+17

7.91725E-26

3.48075E-43

0.71

0.014524092

2.70839E+17

9.44793E-26

3.48839E-43

0.785398163

0.016066598

3.08661E+17

1.07922E-25

3.49645E-43

0.81

0.016569914

3.19997E+17

1.11975E-25

3.49926E-43

0.84

0.017183674

3.32973E+17

1.16634E-25

3.5028E-43

0.87

0.01779744

3.44887E+17

1.20934E-25

3.50648E-43

0.90.018411213

3.556E+17

1.24826E-25

3.51029E-43

0.93

0.019024993

3.64974E+17

1.2826E-25

3.51424E-43

0.935

0.019127291

3.66396E+17

1.28785E-25

3.51491E-43

0.94

0.019229588

3.67776E+17

1.29295E-25

3.51558E-43

0.945

0.019331886

3.69114E+17

1.2979E-25

3.51626E-43

0.95

0.019434184

3.70409E+17

1.30271E-25

3.51694E-43

0.955

0.019536482

3.7166E+17

1.30736E-25

3.51763E-43

0.95510.019538528

3.71685E+17

1.30745E-25

3.51764E-43

0.95520.019540574

3.7171E+17

1.30755E-25

3.51766E-43

Θe0.955316618

0.01954296

3.71738E+17

1.30765E-25

3.51767E-43

Θe×1.00000000000001

0.955316618

0.01954296

3.71738E+17

1.30765E-25

3.51767E-43

0.96

0.01963878

3.72867E+17

1.31187E-25

3.51832E-43

0.97

0.019843378

3.75145E+17

1.3204E-25

3.51971E-43

0.98

0.020047976

3.77238E+17

1.3283E-25

3.52111E-43

0.99

0.020252575

3.79141E+17

1.33554E-25

3.52253E-43

10.020457175

3.80848E+17

1.3421E-25

3.52397E-43

1.047197551

0.021422848

3.86112E+17

1.36334E-25

3.53094E-43

1
π/3

B

π/2×0.999999999890292

1.07

0.021889398

3.86894E+17

1.36745E-25

3.53443E-43

1.1

0.022503222

3.86044E+17

1.36626E-25

3.53914E-43

1.12

0.022912442

3.8423E+17

1.36108E-25

3.54236E-43

1.15

0.02352628

3.79539E+17

1.34634E-25

3.5473E-43

1.170019129

0.023935902

3.75038E+17

1.33164E-25

3.55067E-43

1.27

0.025981723

3.34736E+17

1.19449E-25

3.56844E-43

1.351

0.027639234

2.78149E+17

9.96877E-26

3.58397E-43

1.37

0.028028044

2.61513E+17

9.38247E-26

3.58777E-43

1.47

0.030074483

1.51461E+17

5.46574E-26

3.60868E-43

1.48

0.030279134

1.38299E+17

4.9938E-26

3.61086E-43

1.49

0.030483785

1.24734E+17

4.50671E-26

3.61305E-43

1.50.030688439

1.10762E+17

4.00434E-26

3.61527E-43

1.51

0.030893093

9.63808E+16

3.48657E-26

3.61749E-43

1.52

0.031097749

8.15877E+16

2.95326E-26

3.61974E-43

1.53

0.031302406

6.63804E+16

2.4043E-26

3.622E-43

1.54

0.031507064

5.07565E+16

1.83956E-26

3.62427E-43

1.55

0.031711724

3.47138E+16

1.25892E-26

3.62657E-43

1.56

0.031916385

1.82499E+16

6.62268E-27

3.62888E-43

1.570796327

0.032137345

295252006.8

1.07217E-34

3.63139E-43

π/2×0.999999999996476

1.570796327

0.032137345

9483457.519

3.44381E-36

3.63139E-43

π/2×0.999999999999999

1.570796327

0.032137345

2767.960723

1.00515E-39

3.63139E-43

π/2

1.570796327

0.032137345

485.382281

1.76261E-40

3.63139E-43

case ⑥(wτ=48.901915)

ωτ= wτsinθ

value of [ωτ]0[c]

ω2exp(+ρ)cosδ/(σ 2+ω2)=[ ]2

θ=asin(ωτ/wτ)ω2exp(-ρ)cosδ/(σ 2+ω2)=

ratio/c=3.51767E-43

2sin θexp(wτcosθ)cos(wτsinθ)

sin θexp(-wτcosθ)cos(wτsinθ)

0

0

0

0

π/2×0.00000000000553261063

8.6906E-12

1.77715E-13

5.4611E-05

1.82649E-47#DIV/0!
3.34454E-43

π/2×0.00001312066853

2.06099E-05

4.21454E-07

307136752.5

1.02723E-34

3.34454E-43

0.001

0.001

2.04491E-05

7.23069E+112.41833E-31

3.34454E-43

0.001001

0.001001

2.04695E-05

7.24516E+11

2.42317E-31

3.34454E-43

0.001005

0.001005

2.05513E-05

7.30318E+11

2.44258E-31

3.34454E-43

0.00101

0.00101

2.06536E-05

7.37603E+11

2.46694E-31

3.34454E-43

0.001020.001022.08581E-05

7.52281E+11

2.51603E-31

3.34454E-43

0.00105

0.00105

2.14716E-05

7.97184E+11

2.66621E-31

3.34454E-43

0.00110.00112.2494E-05

8.74914E+11

2.92618E-31

3.34454E-43

0.00120.00122.45389E-05

1.04122E+12

3.4824E-31

3.34454E-43

0.00150.0015

3.06736E-05

1.6269E+12

5.44125E-31

3.34454E-43

0.002

0.002

4.08982E-05

2.89227E+12

9.67332E-31

3.34454E-43

0.003

0.003

6.13473E-05

6.5076E+12

2.17649E-30

3.34454E-43

0.004

0.004

8.17964E-05

1.1569E+13

3.86931E-30

3.34454E-43

0.005

0.005

0.000102245

1.80765E+13

6.04576E-30

3.34454E-43

0.01

0.01

0.000204491

7.23033E+13

2.41822E-29

3.34455E-43
3.34471E-43

0.05
π/32

π/16

π/8

π/4

0.05

0.001022455

1.80537E+15

6.03844E-28

0.09817477

0.002007587

6.93491E+15

2.31987E-27

3.3452E-43

0.120.002453894

1.03358E+16

3.45787E-27

3.34552E-43

0.15

0.003067369

1.60827E+16

5.38139E-27

3.34608E-43

0.17

0.003476353

2.05894E+16

6.89028E-27

3.34652E-43

0.196349541

0.004015182

2.73302E+16

9.1479E-27

3.34718E-43

0.25

0.005112297

4.3759E+16

1.46541E-26

3.34882E-43

0.3

0.006134768

6.21125E+16

2.08121E-26

3.3507E-43

0.35

0.007157245

8.31018E+16

2.78634E-26

3.35293E-43

0.392699082

0.008030428

1.02856E+17

3.45093E-26

3.3551E-43

0.47

0.009611224

1.42085E+17

4.77362E-26

3.35968E-43

0.55

0.01124724

1.85896E+17

6.25593E-26

3.36529E-43

0.63

0.012883287

2.30954E+17

7.78729E-26

3.3718E-43

0.71

0.014519369

2.75001E+17

9.29283E-26

3.3792E-43

0.785398163

0.016061374

3.13405E+17

1.0615E-25

3.387E-43

0.81

0.016564526

3.24915E+17

1.10137E-25

3.38972E-43

0.840.017178086

3.3809E+17

1.14719E-25

3.39315E-43

0.87

0.017791653

3.50187E+17

1.18949E-25

3.39671E-43

0.90.018405226

3.61066E+17

1.22777E-25

3.4004E-43

0.93

0.019018807

3.70583E+17

1.26155E-25

3.40422E-43

0.935

0.019121071

3.72027E+17

1.26671E-25

3.40487E-43

0.94

0.019223335

3.73429E+17

1.27172E-25

3.40553E-43

0.945

0.0193256

3.74787E+17

1.27659E-25

3.40618E-43

0.95

0.019427864

3.76102E+17

1.28132E-25

3.40684E-43

0.955

0.019530129

3.77373E+17

1.2859E-25

3.40751E-43

0.95510.019532174

3.77398E+17

1.28599E-25

3.40752E-43

0.95520.01953422

3.77423E+17

1.28608E-25

3.40753E-43

Θe0.955316618

0.019536605

3.77452E+17

1.28618E-25

3.40755E-43

Θe×1.00000000000001

0.955316618

0.019536605

3.77452E+17

1.28618E-25

3.40755E-43

0.96

0.019632394

3.78598E+17

1.29033E-25

3.40817E-43

1
π/3

B

0.97

0.019836925

3.80911E+17

1.29872E-25

3.40952E-43

0.98

0.020041457

3.83036E+17

1.30649E-25

3.41088E-43

0.99

0.020245989

3.84968E+17

1.31361E-25

3.41225E-43

10.020450522

3.86702E+17

1.32006E-25

3.41364E-43

1.047197551

0.021415881

3.92046E+17

1.34095E-25

3.4204E-43

1.07

0.02188228

3.92841E+17

1.345E-25

3.42378E-43

1.1

0.022495904

3.91978E+17

1.34383E-25

3.42834E-43

1.12

0.022904991

3.90135E+17

1.33873E-25

3.43145E-43
3.43624E-43

1.15

0.02351863

3.85373E+17

1.32423E-25

1.170019129

0.023928118

3.80803E+17

1.30977E-25

3.4395E-43

1.27

0.025973273

3.39882E+17

1.17487E-25

3.45671E-43

1.351

0.027630245

2.82425E+17

9.80509E-26

3.47175E-43

1.37

0.028018929

2.65533E+17

9.22841E-26

3.47543E-43

1.47

0.030064701

1.53789E+17

5.37598E-26

3.49568E-43

1.48

0.030269286

1.40426E+17

4.91179E-26

3.49779E-43

1.49

0.030473871

1.26652E+17

4.4327E-26

3.49991E-43

1.50.030678457

1.12465E+17

3.93858E-26

3.50206E-43

1.51

0.030883045

9.78626E+16

3.42931E-26

3.50421E-43
3.50639E-43

1.52

0.031087634

8.28421E+16

2.90476E-26

1.53

0.031292225

6.7401E+16

2.36481E-26

3.50858E-43

1.54

0.031496817

5.15369E+16

1.80935E-26

3.51078E-43

1.55

0.03170141

3.52475E+16

1.23824E-26

3.513E-43

1.56

0.031906004

1.85305E+16

6.51392E-27

3.51524E-43

π/2×0.999999999890292

1.570796327

0.032126893

299791839.4

1.05457E-34

3.51767E-43

π/2×0.999999999996476

1.570796327

0.032126893

9630036.525

3.38753E-36

3.51767E-43

π/2×0.999999999999999

1.570796327

0.032126893

2810.517619

9.88647E-40

3.51767E-43

π/2

1.570796327

0.032126893

492.844946

1.73366E-40

3.51767E-43

case ⑦(wτ=48.87668)

ωτ= wτsinθ

value of [ωτ]0[c]

θ=asin(ωτ/wτ)exp(+ρ)=exp(wτcosθ)[ ]exp(-ρ)=ratio/c=3.51767E-43

exp(-wτcosθ)001.68606E+21

5.931E-22

3.51767E-43

π/2×0.00000000000553261063

8.6906E-12

1.77807E-13

1.68606E+21

5.931E-22

3.51767E-43

π/2×0.000013120 66853

2.06099E-05

4.21671E-07

1.68606E+21

5.931E-22

3.51767E-43

0.001

0.001

2.04597E-05

1.68606E+21

5.931E-22

3.51767E-43

0.001001

0.001001

2.04801E-05

1.68606E+21

5.931E-22

3.51767E-43

0.001005

0.001005

2.0562E-05

1.68606E+21

5.931E-22

3.51767E-43

0.00101

0.00101

2.06643E-05

1.68606E+21

5.931E-22

3.51767E-43

0.001020.001022.08688E-05

1.68606E+21

5.931E-22

3.51767E-43

0.00105

0.00105

2.14826E-05

1.68606E+21

5.931E-22

3.51767E-43

0.00110.00112.25056E-05

1.68606E+21

5.931E-22

3.51767E-43

0.00120.00122.45516E-05

1.68606E+21

5.931E-22

3.51767E-43

0.00150.00153.06895E-05

1.68606E+21

5.931E-22

3.51767E-43

0.002

0.002

4.09193E-05

1.68606E+21

5.931E-22

3.51767E-43

0.003

0.003

6.1379E-05

1.68606E+21

5.931E-22

3.51767E-43

0.004

0.004

8.18386E-05

1.68606E+21

5.931E-22

3.51767E-43

0.005

0.005

0.000102298

1.68606E+21

5.931E-22

3.51767E-43

0.01

0.01

0.000204597

1.68606E+21

5.931E-22

3.51768E-43

0.05

0.05

0.001022983

1.68601E+21

5.93115E-22

3.51785E-43

0.09817477

0.002008623

1.68589E+21

5.93158E-22

3.51836E-43

0.12

0.002455161

1.68581E+21

5.93187E-22

3.51871E-43

0.15

0.003068953

1.68567E+21

5.93236E-22

3.51929E-43

0.17

0.003478148

1.68556E+21

5.93275E-22

3.51975E-43

0.196349541

0.004017255

1.68539E+21

5.93333E-22

3.52045E-43

0.25

0.005114936

1.68498E+21

5.93479E-22

3.52217E-43

0.3

0.006137935

1.68451E+21

5.93646E-22

3.52415E-43

0.35

0.00716094

1.68395E+21

5.93843E-22

3.5265E-43

0.392699082

0.008034574

1.6834E+21

5.94036E-22

3.52879E-43

0.47

0.009616186

1.68225E+21

5.94441E-22

3.53361E-43

0.55

0.011253048

1.68085E+21

5.94938E-22

3.53951E-43

0.63

0.012889939

1.67923E+21

5.95513E-22

3.54635E-43

0.71

0.014526866

1.67738E+21

5.96166E-22

3.55414E-43

0.785398163

0.016069667

1.67545E+21

5.96854E-22

3.56235E-43

0.81

0.016573079

1.67478E+21

5.97094E-22

3.56521E-43

0.84

0.017186956

1.67393E+21

5.97396E-22

3.56882E-43

0.87

0.01780084

1.67305E+21

5.9771E-22

3.57257E-43

0.90.01841473

1.67214E+21

5.98035E-22

3.57646E-43

0.93

0.019028627

1.6712E+21

5.98371E-22

3.58048E-43

0.935

0.019130944

1.67104E+21

5.98428E-22

3.58116E-43

0.94

0.019233261

1.67088E+21

5.98485E-22

3.58185E-43

0.945

0.019335579

1.67072E+21

5.98543E-22

3.58254E-43

0.95

0.019437896

1.67056E+21

5.98601E-22

3.58323E-43

0.955

0.019540214

1.6704E+21

5.98659E-22

3.58393E-43

0.95510.01954226

1.6704E+21

5.98661E-22

3.58395E-43

0.95520.019544306

1.67039E+21

5.98662E-22

3.58396E-43

Θe0.955316618

0.019546693

1.67039E+21

5.98663E-22

3.58398E-43

Θe×1.00000000000001

0.955316618

0.019546693

1.67039E+21

5.98663E-22

3.58398E-43

0.96

0.019642532

1.67023E+21

5.98718E-22

3.58463E-43

0.97

0.019847168

1.66991E+21

5.98836E-22

3.58605E-43

0.98

0.020051805

1.66957E+21

5.98956E-22

3.58748E-43

0.99

0.020256443

1.66924E+21

5.99077E-22

3.58893E-43

10.020461082

1.6689E+21

5.99199E-22

3.59039E-43

1.047197551

0.02142694

1.66725E+21

5.99791E-22

3.5975E-43

1.07

0.02189358

1.66642E+21

6.00088E-22

3.60105E-43

π/32

π/16

π/8

π/4

1
π/3

1.1

0.022507521

1.66531E+21

6.00488E-22

3.60585E-43

1.12

0.022916819

1.66456E+21

6.0076E-22

3.60913E-43

1.15

0.023530774

1.6634E+21

6.01179E-22

3.61416E-43

1.170019129

0.023940474

1.66261E+21

6.01465E-22

3.6176E-43

1.27

0.025986686

1.65846E+21

6.02968E-22

3.63571E-43

1.351

0.027644515

1.65486E+21

6.0428E-22

3.65154E-43

1.37

0.028033399

1.65399E+21

6.046E-22

3.65541E-43

1.47

0.030080229

1.64919E+21

6.06359E-22

3.67672E-43

1.48

0.030284918

1.64869E+21

6.06542E-22

3.67894E-43

1.49

0.030489609

1.64819E+21

6.06727E-22

3.68118E-43

1.50.030694302

1.64768E+21

6.06913E-22

3.68343E-43

1.51

0.030898995

1.64718E+21

6.071E-22

3.6857E-43

1.52

0.03110369

1.64667E+21

6.07288E-22

3.68799E-43

1.53

0.031308386

1.64615E+21

6.07477E-22

3.69029E-43

1.54

0.031513084

1.64563E+21

6.07668E-22

3.69261E-43

1.55

0.031717783

1.64511E+21

6.07861E-22

3.69495E-43

1.56

0.031922483

1.64459E+21

6.08054E-22

3.6973E-43

π/2×0.999999999890292

1.570796327

0.032143485

1.64402E+21

6.08265E-22

3.69986E-43

π/2×0.999999999996476

1.570796327

0.032143485

1.64402E+21

6.08265E-22

3.69986E-43

π/2

1.570796327

0.032143485

1.64402E+21

6.08265E-22

3.69986E-43

Bcase ⑧(wτ=48.886015)

ωτ= wτsinθ

value of [ωτ]0[c]

θ=asin(ωτ/wτ)exp(+ρ)=exp(wτcosθ)[ ]exp(-ρ)=ratio/c=3.51767E-43

exp(-wτcosθ)001.70187E+21

5.87589E-22

3.4526E-43

π/2×0.00000000000553261063

8.6906E-12

1.77773E-13

1.70187E+21

5.87589E-22

3.4526E-43

π/2×0.000013120 66853

2.06099E-05

4.21591E-07

1.70187E+21

5.87589E-22

3.4526E-43

0.001

2.04557E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.001
0.001001

0.001001

2.04762E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.001005

0.001005

2.0558E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.00101

0.00101

2.06603E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.001020.001022.08649E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.00105

0.00105

2.14785E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.00110.00112.25013E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.00120.00122.45469E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.00150.00153.06836E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.002

0.002

4.09115E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.003

0.003

6.13672E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.004

0.004

8.1823E-05

1.70187E+21

5.87589E-22

3.45261E-43

0.005

0.005

0.000102279

1.70187E+21

5.87589E-22

3.45261E-43

0.01

0.01

0.000204557

1.70187E+21

5.87589E-22

3.45261E-43

0.05

0.05

0.001022788

1.70183E+21

5.87604E-22

3.45278E-43

0.09817477

0.00200824

1.7017E+21

5.87647E-22

3.45329E-43

0.12

0.002454692

1.70162E+21

5.87675E-22

3.45362E-43

0.15

0.003068367

1.70148E+21

5.87724E-22

3.45419E-43

0.17

0.003477484

1.70137E+21

5.87762E-22

3.45465E-43

0.196349541

0.004016488

1.7012E+21

5.8782E-22

3.45533E-43

0.25

0.005113959

1.70078E+21

5.87964E-22

3.45702E-43

0.3

0.006136763

1.7003E+21

5.8813E-22

3.45897E-43

0.35

0.007159573

1.69974E+21

5.88325E-22

3.46127E-43

0.392699082

0.00803304

1.69919E+21

5.88516E-22

3.46351E-43

0.47

0.00961435

1.69803E+21

5.88918E-22

3.46824E-43

0.55

0.011250899

1.69661E+21

5.8941E-22

3.47404E-43

0.63

0.012887478

1.69498E+21

5.89979E-22

3.48075E-43

0.71

0.014524092

1.69312E+21

5.90626E-22

3.48839E-43

0.785398163

0.016066598

1.69117E+21

5.91308E-22

3.49645E-43

0.81

0.016569914

1.69049E+21

5.91545E-22

3.49926E-43

0.84

0.017183674

1.68963E+21

5.91845E-22

3.5028E-43

0.87

0.01779744

1.68875E+21

5.92156E-22

3.50648E-43

π/32

π/16

π/8

π/4

0.9

0.018411213

1.68783E+21

5.92477E-22

3.51029E-43

0.93

0.019024993

1.68688E+21

5.9281E-22

3.51424E-43

0.935

0.019127291

1.68672E+21

5.92867E-22

3.51491E-43

0.94

0.019229588

1.68656E+21

5.92924E-22

3.51558E-43

0.945

0.019331886

1.6864E+21

5.92981E-22

3.51626E-43

0.95

0.019434184

1.68623E+21

5.93038E-22

3.51694E-43

0.955

0.019536482

1.68607E+21

5.93096E-22

3.51763E-43

0.95510.019538528

1.68606E+21

5.93097E-22

3.51764E-43

0.95520.019540574

1.68606E+21

5.93098E-22

3.51766E-43

Θe0.955316618

0.01954296

1.68606E+21

5.931E-22

3.51767E-43

Θe×1.00000000000001

0.955316618

0.01954296

1.68606E+21

5.931E-22

3.51767E-43

0.96

0.01963878

1.6859E+21

5.93154E-22

3.51832E-43

0.97

0.019843378

1.68557E+21

5.93271E-22

3.51971E-43

0.98

0.020047976

1.68523E+21

5.9339E-22

3.52111E-43

0.99

0.020252575

1.68489E+21

5.93509E-22

3.52253E-43

10.020457175

1.68455E+21

5.9363E-22

3.52397E-43

1.047197551

0.021422848

1.68289E+21

5.94217E-22

3.53094E-43

1.07

0.021889398

1.68206E+21

5.94511E-22

3.53443E-43

1.1

0.022503222

1.68094E+21

5.94907E-22

3.53914E-43

1.12

0.022912442

1.68017E+21

5.95177E-22

3.54236E-43

1.15

0.02352628

1.679E+21

5.95592E-22

3.5473E-43

1.170019129

0.023935902

1.6782E+21

5.95875E-22

3.55067E-43

1
π/3

B

1.27

0.025981723

1.67402E+21

5.97364E-22

3.56844E-43

1.351

0.027639234

1.67039E+21

5.98663E-22

3.58397E-43

1.37

0.028028044

1.66951E+21

5.9898E-22

3.58777E-43

1.47

0.030074483

1.66466E+21

6.00723E-22

3.60868E-43

1.48

0.030279134

1.66416E+21

6.00904E-22

3.61086E-43

1.49

0.030483785

1.66365E+21

6.01087E-22

3.61305E-43

1.50.030688439

1.66314E+21

6.01271E-22

3.61527E-43

1.51

0.030893093

1.66263E+21

6.01456E-22

3.61749E-43

1.52

0.031097749

1.66212E+21

6.01643E-22

3.61974E-43

1.53

0.031302406

1.6616E+21

6.0183E-22

3.622E-43

1.54

0.031507064

1.66108E+21

6.02019E-22

3.62427E-43

1.55

0.031711724

1.66055E+21

6.0221E-22

3.62657E-43

1.56

0.031916385

1.66002E+21

6.02401E-22

3.62888E-43

π/2×0.999999999890292

1.570796327

0.032137345

1.65945E+21

6.0261E-22

3.63139E-43

π/2×0.999999999996476

1.570796327

0.032137345

1.65945E+21

6.0261E-22

3.63139E-43

π/2

1.570796327

0.032137345

1.65945E+21

6.0261E-22

3.63139E-43

case ⑨(wτ=48.901915)

ωτ= wτsinθ

value of [ωτ]0[c]

θ=asin(ωτ/wτ)exp(+ρ)=exp(wτcosθ)[ ]exp(-ρ)=ratio/c=3.51767E-43

exp(-wτcosθ)

product c*case ⑦,⑧,⑨

0

01.72915E+21

5.7832E-22

3.34454E-43

1π/2×0.00000000000553261063

8.6906E-12

1.77715E-13

1.72915E+21

5.7832E-22

3.34454E-43

1π/2×0.000013120 66853

2.06099E-05

4.21454E-07

1.72915E+21

5.7832E-22

3.34454E-43

1

0.001

0.001

2.04491E-05

1.72915E+21

5.7832E-22

3.34454E-43

10.001001

0.001001

2.04695E-05

1.72915E+21

5.7832E-22

3.34454E-43

10.001005

0.001005

2.05513E-05

1.72915E+21

5.7832E-22

3.34454E-43

10.00101

0.00101

2.06536E-05

1.72915E+21

5.7832E-22

3.34454E-43

10.001020.001022.08581E-05

1.72915E+21

5.7832E-22

3.34454E-43

10.00105

0.00105

2.14716E-05

1.72915E+21

5.7832E-22

3.34454E-43

10.00110.00112.2494E-05

1.72915E+21

5.7832E-22

3.34454E-43

10.00120.00122.45389E-05

1.72915E+21

5.7832E-22

3.34454E-43

10.00150.0015

3.06736E-05

1.72915E+21

5.7832E-22

3.34454E-43

1

0.002

0.002

4.08982E-05

1.72915E+21

5.7832E-22

3.34454E-43

1

0.003

0.003

6.13473E-05

1.72915E+21

5.7832E-22

3.34454E-43

1

0.004

0.004

8.17964E-05

1.72915E+21

5.7832E-22

3.34454E-43

1

0.005

0.005

0.000102245

1.72915E+21

5.7832E-22

3.34454E-43

1

0.01

0.01

0.000204491

1.72914E+21

5.78321E-22

3.34455E-43

1

0.05

0.05

0.001022455

1.7291E+21

5.78335E-22

3.34471E-43

10.09817477

0.002007587

1.72898E+21

5.78377E-22

3.3452E-43

1

0.12

0.002453894

1.72889E+21

5.78405E-22

3.34552E-43

1

0.15

0.003067369

1.72875E+21

5.78453E-22

3.34608E-43

1

0.17

0.003476353

1.72864E+21

5.78491E-22

3.34652E-43

10.196349541

0.004015182

1.72847E+21

5.78548E-22

3.34718E-43

1

0.25

0.005112297

1.72804E+21

5.7869E-22

3.34882E-43

10.3

0.006134768

1.72756E+21

5.78852E-22

3.3507E-43

1

0.35

0.007157245

1.72698E+21

5.79045E-22

3.35293E-43

10.392699082

0.008030428

1.72642E+21

5.79233E-22

3.3551E-43

1

0.47

0.009611224

1.72525E+21

5.79628E-22

3.35968E-43

1

0.55

0.01124724

1.72381E+21

5.80111E-22

3.36529E-43

1

0.63

0.012883287

1.72214E+21

5.80672E-22

3.3718E-43

1

0.71

0.014519369

1.72026E+21

5.81309E-22

3.3792E-43

10.785398163

0.016061374

1.71827E+21

5.81979E-22

3.387E-43

1

0.81

0.016564526

1.71759E+21

5.82213E-22

3.38972E-43

1

0.84

0.017178086

1.71672E+21

5.82508E-22

3.39315E-43

1

0.87

0.017791653

1.71582E+21

5.82813E-22

3.39671E-43

10.9

0.018405226

1.71488E+21

5.8313E-22

3.4004E-43

1

π/32

π/16

π/8

π/4

0.93

0.019018807

1.71392E+21

5.83457E-22

3.40422E-43

1

0.935

0.019121071

1.71376E+21

5.83513E-22

3.40487E-43

1

0.94

0.019223335

1.71359E+21

5.83569E-22

3.40553E-43

1

0.945

0.0193256

1.71343E+21

5.83625E-22

3.40618E-43

1

0.95

0.019427864

1.71326E+21

5.83682E-22

3.40684E-43

1

0.955

0.019530129

1.7131E+21

5.83739E-22

3.40751E-43

10.95510.019532174

1.71309E+21

5.8374E-22

3.40752E-43

10.95520.01953422

1.71309E+21

5.83741E-22

3.40753E-431Θe0.955316618

0.019536605

1.71309E+21

5.83742E-22

3.40755E-43

1Θe×1.00000000000001

0.955316618

0.019536605

1.71309E+21

5.83742E-22

3.40755E-43

1

0.96

0.019632394

1.71293E+21

5.83796E-22

3.40817E-43

1

0.97

0.019836925

1.71259E+21

5.83911E-22

3.40952E-43

1

0.98

0.020041457

1.71225E+21

5.84027E-22

3.41088E-43

1

0.99

0.020245989

1.7119E+21

5.84145E-22

3.41225E-43

1

10.020450522

1.71156E+21

5.84264E-22

3.41364E-43

1

1.047197551

0.021415881

1.70986E+21

5.84842E-22

3.4204E-43

1

1.07

0.02188228

1.70902E+21

5.8513E-22

3.42378E-43

1

1.1

0.022495904

1.70788E+21

5.8552E-22

3.42834E-43

1

1.12

0.022904991

1.70711E+21

5.85786E-22

3.43145E-43

1

1.15

0.02351863

1.70592E+21

5.86194E-22

3.43624E-43

1

1.170019129

0.023928118

1.70511E+21

5.86473E-22

3.4395E-43

1

1.27

0.025973273

1.70086E+21

5.87938E-22

3.45671E-43

1

1.351

0.027630245

1.69717E+21

5.89216E-22

3.47175E-43

1

1.37

0.028018929

1.69627E+21

5.89528E-22

3.47543E-43

1

1.47

0.030064701

1.69135E+21

5.91243E-22

3.49568E-43

1

1.48

0.030269286

1.69084E+21

5.91421E-22

3.49779E-43

1

1.49

0.030473871

1.69033E+21

5.91601E-22

3.49991E-43

11.5

0.030678457

1.68981E+21

5.91782E-22

3.50206E-43

1

1.51

0.030883045

1.68929E+21

5.91964E-22

3.50421E-43

1

1.52

0.031087634

1.68877E+21

5.92147E-22

3.50639E-43

1

1.53

0.031292225

1.68824E+21

5.92332E-22

3.50858E-43

1

1.54

0.031496817

1.68771E+21

5.92518E-22

3.51078E-43

1

1.55

0.03170141

1.68718E+21

5.92706E-22

3.513E-43

11.56

0.031906004

1.68664E+21

5.92894E-22

3.51524E-43

1π/2×0.999999999890292

1.570796327

0.032126893

1.68606E+21

5.93099E-22

3.51767E-43

1π/2×0.999999999996476

1.570796327

0.032126893

1.68606E+21

5.93099E-22

3.51767E-431π/2

1.570796327

0.032126893

1.68606E+21

5.93099E-22

3.51767E-43

1

1
π/3

B

[VAMS] unit system
Physical quantity
LengthTime
Mass
Momentum
Force
EnergyAngular momentum
PowerElectric charge, Electric flux
Magnetic flux, Magnetic charge
Electric potential, Magnetic current
Electric current, Magnetic potential
Electric current density
Density of electric charge
Electric displacement
Flux density
Electric field
Magnetic field
Dielectric constant
Magnetic permeability
Electric resistance
ConductanceElectric conductivity
Electric capacity
InductanceElectric moment
Magnetic moment
Electric polarization
Magnetic polarization

Power of Dimention[VAMS] unit system[SI] unit system
V
A
m
s
A
Kg
m
s
m
0
0
1
0
0
0
1
0
s
0
0
0
1
0
0
0
1
Kg
1
1
-2
3
0
1
0
0
N・s
1
1
-1
2
0
1
1
-1
N
1
1
-1
1
0
1
1
-2
J
1
1
0
1
0
1
2
-2
J・s
1
1
0
2
0
1
2
-1
P
1
1
0
0
0
1
2
-3
C
0
1
0
1
1
0
0
1
wb
1
0
0
1
-1
1
2
-2
V
1
0
0
0
-1
1
2
-3
A
0
1
0
0
1
0
0
0
A/Am2
0
1
-2
0
1
0
-2
0
C/m3
0
1
-3
1
1
0
-3
1
C/m2
0
1
-2
1
1
0
-2
1
T
1
0
-2
1
-1
1
0
-2
V/m
1
0
-1
0
-1
1
1
-3
A/m
0
1
-1
0
1
0
-1
0
F/m
-1
1
-1
1
2
-1
-3
4
H/m
1
-1
-1
1
-2
1
1
-2
Ω
1
-1
0
0
-2
1
2
-3
S
-1
1
0
0
2
-1
-2
3
1/(Ω・m) -1
1
-1
0
2
-1
-3
3
F
-1
1
0
1
2
-1
-2
4
H
1
-1
0
1
-2
1
2
-2
C・m
0
1
1
1
1
0
1
1
Wb・m
1
0
1
1
-1
1
3
-2
C/m2
0
1
-2
1
1
0
-2
1
Wb/m2
1
0
-2
1
-1
1
0
-2
Unit

[VAMS] 単位系
物理量
長さ
時間
質量
運動量
力
エネルギー
角運動量
仕事率
電荷、電束
磁束、磁荷
電位、磁流
電流、磁位
電流密度
電荷密度
電束密度
磁束密度
電場
磁場
誘電率
透磁率
電気抵抗
コンダクタンス
電気伝導度
電気容量
インダクタンス
電気モーメント
磁気モーメント
電気分極
磁気分極

基本単位ベキ[VAMS]単位系[SI]単位系
V
A
m
s
A
Kg
m
m
0
0
1
0
0
0
1
s
0
0
0
1
0
0
0
Kg
1
1
-2
3
0
1
0
N・s
1
1
-1
2
0
1
1
N
1
1
-1
1
0
1
1
J
1
1
0
1
0
1
2
J・s
1
1
0
2
0
1
2
P
1
1
0
0
0
1
2
C
0
1
0
1
1
0
0
wb
1
0
0
1
-1
1
2
V
1
0
0
0
-1
1
2
A
0
1
0
0
1
0
0
2
A/Am
0
1
-2
0
1
0
-2
C/m3
0
1
-3
1
1
0
-3
C/m2
0
1
-2
1
1
0
-2
T
1
0
-2
1
-1
1
0
V/m
1
0
-1
0
-1
1
1
A/m
0
1
-1
0
1
0
-1
F/m
-1
1
-1
1
2
-1
-3
H/m
1
-1
-1
1
-2
1
1
Ω
1
-1
0
0
-2
1
2
S
-1
1
0
0
2
-1
-2
1/(Ω・m) -1
1
-1
0
2
-1
-3
F
-1
1
0
1
2
-1
-2
H
1
-1
0
1
-2
1
2
C・m
0
1
1
1
1
0
1
Wb・m
1
0
1
1
-1
1
3
2
C/m
0
1
-2
1
1
0
-2
2
Wb/m
1
0
-2
1
-1
1
0
単位

s
0
1
0
-1
-2
-2
-1
-3
1
-2
-3
0
0
1
1
-2
-3
0
4
-2
-3
3
3
4
-2
1
-2
1
-2