aarjaneiro/planetmath huggingface dataset:
quasiperiodic function

quasiperiodic function

A

function

f

is said to have a

quasiperiod

p

if there exists a function

g

such that

f

⁢

(

z

+

p

)

=

g

⁢

(

z

)

⁢

f

⁢

(

z

)

.

In the special case where

g

is identically equal to

1

, we call

f

a

periodic function

, and we say that

p

is a

period

of

f

or that

f

has

periodicity

p

.

Except for the special case of periodicity noted above, the notion of quasiperiodicity is somewhat loose and fuzzy. Strictly speaking, many functions could be regarded as quasiperiodic if one defines

g

⁢

(

z

)

=

f

⁢

(

z

+

p

)

/

f

⁢

(

z

)

. In order for the term “quasiperiodic” not to be trivial, it is customary to reserve its use for the case where the function

g

is, in some vague, intuitive sense, simpler than the function

f

. For instance, no one would call the function

f

⁢

(

z

)

=

z

2

+

1

quasiperiodic even though it meets the criterion of the definition if we set

g

⁢

(

z

)

=

(

z

2

+

2

⁢

z

+

2

)

/

(

z

2

+

1

)

because the rational function

g

is “more complicated” than the

polynomial

f

. On the other hand, for the

gamma function

, one would say that

1

is a quasiperiod because

Γ

⁢

(

z

+

1

)

=

z

⁢

Γ

⁢

(

z

)

and the function

g

⁢

(

z

)

=

z

is a “much simpler” function than the gamma function.

Note that the every

complex number

can be said to be a quasiperiod of the

exponential function

. The term “quasiperiod” is most frequently used in connection with

theta functions

.

Also note that almost periodic functions are quite a different affair than quasiperiodic functions — there, one is dealing with a precise notion.

Title

quasiperiodic function

Canonical name

QuasiperiodicFunction

Date of creation

2013-03-22 14:40:38

Last modified on

2013-03-22 14:40:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 30A99

Related topic

ComplexTangentAndCotangent

Related topic

CounterperiodicFunction

Defines

quasiperiod

Defines

quasiperiodicity

Defines

period

Defines

periodic function

Defines

periodic

Defines

periodicity

Generated on Fri Feb 9 20:14:45 2018 by

LaTeXML
Riemann mapping theorem

Riemann mapping theorem

Let

U

be a simply connected open

proper subset

of

ℂ

, and let

a

∈

U

. There is a unique

analytic function

f

:

U

→

ℂ

such that

1.

f

⁢

(

a

)

=

0

, and

f

′

⁢

(

a

)

is real and positive;

2.

f

is

injective

;

3.

f

⁢

(

U

)

=

{

z

∈

ℂ

:

|

z

|

<

1

}

.

Remark.

As a

consequence

of this

theorem

, any two simply connected regions, none of which is the whole plane, are

conformally equivalent

.

Title

Riemann mapping theorem

Canonical name

RiemannMappingTheorem

Date of creation

2013-03-22 13:15:03

Last modified on

2013-03-22 13:15:03

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Theorem

Classification

msc 30A99

Related topic

ConformalRadius

Generated on Fri Feb 9 20:14:47 2018 by

LaTeXML
sense-preserving mapping

sense-preserving mapping

A

continuous mapping

which

preserves

the

orientation

of a

Jordan curve

is called

sense-preserving

or

orientation-preserving

. If on the other hand a mapping reverses the orientation, it is called sense-reversing.

If the mapping is furthermore

differentiable

then the above statement is equivalent to saying that the

Jacobian

is

strictly

positive

at every point of the domain.

An example of sense-preserving mapping is any

conformal mapping

f

:

ℂ

→

ℂ

. If you however look at the mapping

g

⁢

(

z

)

:=

f

⁢

(

z

¯

)

, then that is a sense-reversing mapping. In general if

f

:

ℂ

→

ℂ

is a smooth mapping then the Jacobian in fact is defined as

J

=

|

f

z

|

-

|

f

z

¯

|

, and so a mapping is sense preserving if the

modulus

of the

partial derivative

with respect to

z

is strictly

greater

then the modulus of the partial derivative with respect to

z

¯

.

This does not that this notion is to the

complex plane

. For example

f

:

ℝ

→

ℝ

defined by

f

⁢

(

x

)

=

2

⁢

x

is a sense preserving mapping, while

f

⁢

(

x

)

=

x

2

is sense preserving only on the

interval

(

0

,

∞

)

.

Title

sense-preserving mapping

Canonical

name

SensepreservingMapping

Date of creation

2013-03-22 14:08:01

Last modified on

2013-03-22 14:08:01

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 30A99

Classification

msc 26B05

Synonym

orientation-preserving

Related topic

Orientation

Related topic

Jacobian

Related topic

Curve

Defines

sense-preserving

Defines

sense-reversing

Generated on Fri Feb 9 20:14:49 2018 by

LaTeXML
sine integral

sine integral

The

function

sine integral

(in Latin

sinus integralis

) from

ℝ

to

ℝ

is defined as

Si

⁢

x

:=

∫

0

x

sin

⁡

t

t

⁢

𝑑

t

=

∫

0

x

sinc

⁢

(

t

)

⁢

𝑑

t

,

(1)

or alternatively as

Si

⁢

x

:=

∫

0

1

sin

⁡

t

⁢

x

t

⁢

𝑑

t

.

It isn’t an

elementary function

.  The equation (1) implies the

Taylor series

Si

z

=

z

-

z

3

3

⋅

3

!

+

z

5

5

⋅

5

!

-

z

7

7

⋅

7

!

+

-

…

,

which converges for all

complex

values

z

and thus defines an

entire transcendental function

.

Si

⁢

x

satisfies the linear third

differential equation

x

⁢

f

′′′

⁢

(

x

)

+

2

⁢

f

′′

⁢

(

x

)

+

x

⁢

f

′

⁢

(

x

)

=

0

.

Remark 1.

lim

x

→

∞

⁡

Si

⁢

x

=

π

2

Remark 2.

There is also another “sine integral”

si

⁢

x

:=

∫

∞

x

sin

⁡

t

t

⁢

𝑑

t

=

Si

⁢

x

-

π

2

and the corresponding

cosine integral

ci

⁢

x

:=

∫

∞

x

cos

⁡

t

t

⁢

𝑑

t

=

γ

+

ln

⁡

x

+

∫

0

x

cos

⁡

t

-

1

t

⁢

𝑑

t

where

γ

is the Euler–Mascheroni constant (

http://planetmath.org/EulerMascheroniConstant

).

Title

sine integral

Canonical

name

SineIntegral

Date of creation

2015-02-04 12:58:26

Last modified on

2015-02-04 12:58:26

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 30A99

Synonym

sinus integralis

Synonym

Si

Related topic

SincFunction

Related topic

SineIntegralInInfinity

Related topic

LogarithmicIntegral2

Related topic

CurvatureOfNielsensSpiral

Related topic

LaplaceTransformOfIntegralSine

Related topic

FresnelIntegrals

Related topic

HyperbolicSineIntegral

Defines

sine integral

Defines

sinus integralis

Defines

cosine integral

Generated on Fri Feb 9 20:14:51 2018 by

LaTeXML
topic entry on series

topic entry on series

There are in PM many entries on series whose terms are usually thought to be real but which can equally well be any

complex numbers

. It may be good to make a list of such entries, but of course include such ones concerning the proper

complex

(

imaginary

) term series.

1.

series (

http://planetmath.org/Series

)

2.

convergent series

3.

determining series convergence

4.

convergence of complex term series

5.

sum of series

6.

remainder term

7.

finite changes in convergent series

8.

manipulating convergent series

9.

Riemann’s theorem on rearrangements

10.

necessary condition of

convergence

11.

harmonic series

12.

http://planetmath.org/node/11941

convergent series where not only

a

n

but also

n

⁢

a

n

tends to 0

13.

generalization of Olivier criterion (

http://planetmath.org/AGeneralisationOfOlivierCriterion

)

14.

slower convergent series

15.

slower divergent series

16.

adding and removing parentheses in series

17.

absolute convergence theorem

18.

limit of sequence as sum of series

19.

sum of series depends on order

20.

conditionally convergent real series

21.

multiplication of series

22.

Cauchy product

23.

double series

24.

absolute convergence of double series

25.

Dirichlet series

26.

convergence of Riemann zeta series

27.

example of

summation by parts

28.

real part series and imaginary part series

29.

sum

function

of series

30.

termwise differentiation

31.

theorems on complex function series

32.

power series

33.

identity theorem of power series

34.

Taylor series

35.

Laurent series

36.

binomial theorem

37.

quotient of Taylor series

38.

Abel summability

39.

Weierstrass double series theorem

40.

infinity

sum of

reciprocal numbers

Title

topic entry on series

Canonical name

TopicEntryOnSeries

Date of creation

2013-03-22 17:27:59

Last modified on

2013-03-22 17:27:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

25

Author

pahio (2872)

Entry type

Topic

Classification

msc 30A99

Synonym

topic entry on series of complex terms

Generated on Fri Feb 9 20:14:56 2018 by

LaTeXML
unit disk

unit disk

The

unit disk

in the

complex plane

, denoted

Δ

, is defined as

{

z

∈

ℂ

:

|

z

|

<

1

}

. The

unit circle

, denoted

∂

⁡

Δ

or

S

1

is the boundary

{

z

∈

ℂ

:

|

z

|

=

1

}

of the unit disk

Δ

. Every element

z

∈

∂

⁡

Δ

can be written as

z

=

e

i

⁢

θ

for some real value of

θ

.

Title

unit disk

Canonical name

UnitDisk

Date of creation

2013-03-22 13:37:45

Last modified on

2013-03-22 13:37:45

Owner

brianbirgen (2180)

Last modified by

brianbirgen (2180)

Numerical id

8

Author

brianbirgen (2180)

Entry type

Definition

Classification

msc 30A99

Synonym

unit disc

Related topic

ConformalMobiusCircleMapTheorem

Related topic

SchwarzLemma

Related topic

Complex

Related topic

UpperHalfPlane

Related topic

UnitDiskUpperHalfPlaneConformalEquivalenceTheorem

Related topic

UnitDisc2

Defines

unit circle

Generated on Fri Feb 9 20:14:58 2018 by

LaTeXML
upper half plane

upper half plane

The

upper half plane

in the

complex plane

, abbreviated UHP, is defined as

{

z

∈

ℂ

:

Im

⁡

(

z

)

>

0

}

.

Title

upper half plane

Canonical name

UpperHalfPlane

Date of creation

2013-03-22 13:37:49

Last modified on

2013-03-22 13:37:49

Owner

brianbirgen (2180)

Last modified by

brianbirgen (2180)

Numerical id

9

Author

brianbirgen (2180)

Entry type

Definition

Classification

msc 30A99

Related topic

UnitDisk

Related topic

Complex

Related topic

UnitDiskUpperHalfPlaneConformalEquivalenceTheorem

Related topic

ConvertingBetweenThePoincareDiscModelAndTheUpperHalfPlaneModel

Defines

UHP

Generated on Fri Feb 9 20:15:00 2018 by

LaTeXML
Weierstrass M-test

Weierstrass M-test

Let

X

be any set,

{

f

n

}

n

∈

ℕ

a

sequence

of real or

complex

valued

functions

on

X

and

{

M

n

}

n

∈

ℕ

a sequence of non-negative

real numbers

.
Suppose that, for each

n

∈

ℕ

and

x

∈

X

, we have

|

f

n

⁢

(

x

)

|

≤

M

n

. Then

f

=

∑

n

=

1

∞

f

n

converges uniformly

if

∑

n

=

1

∞

M

n

converges

.

Title

Weierstrass M-test

Canonical name

WeierstrassMtest

Date of creation

2013-03-22 12:56:11

Last modified on

2013-03-22 12:56:11

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Theorem

Classification

msc 30A99

Related topic

AbsoluteConvergence

Generated on Fri Feb 9 20:15:01 2018 by

LaTeXML
Weierstrass M-test for continuous functions

Weierstrass M-test for continuous functions

When the set

X

in the statement of the

Weierstrass M-test

is a

topological space

, a strengthening of the

hypothesis

produces a

stronger

result. When the

functions

f

n

are

continuous

, then the limit of the series

f

=

∑

n

=

1

∞

f

n

is also continuous.

The proof follows directly from the fact that the limit of a

uniformly convergent

sequence

of

continuous functions

is continuous.

Title

Weierstrass M-test for continuous functions

Canonical

name

WeierstrassMtestForContinuousFunctions

Date of creation

2013-03-22 16:08:48

Last modified on

2013-03-22 16:08:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Corollary

Classification

msc 30A99

Generated on Fri Feb 9 20:15:03 2018 by

LaTeXML
Weierstrass’ theorem on addition formulas

Weierstrass’ theorem on addition formulas

Suppose that a

complex analytic function

f

has an

addition formula

. Then we can express

f

as a

composition

f

=

g

∘

h

, where

g

is an

algebraic function

and

h

is one of the following:

•

the

identity function

•

the

exponential function

•

an

elliptic function

Title

Weierstrass’ theorem on addition formulas

Canonical name

WeierstrassTheoremOnAdditionFormulas

Date of creation

2013-03-22 14:45:47

Last modified on

2013-03-22 14:45:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 30A99

Classification

msc 30D05

Generated on Fri Feb 9 20:15:05 2018 by

LaTeXML
winding number and fundamental group

winding number and fundamental group

The

winding number

is an

analytic

way to define an explicit

isomorphism

W

[

∙

:

z

0

]

:

π

1

(

ℂ

∖

z

0

)

→

ℤ

from the

fundamental group

of the punctured (at

z

0

)

complex plane

to the group
of integers.

Title

winding number and fundamental group

Canonical name

WindingNumberAndFundamentalGroup

Date of creation

2013-03-22 12:58:12

Last modified on

2013-03-22 12:58:12

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

5

Author

Dr_Absentius (537)

Entry type

Result

Classification

msc 30A99

Classification

msc 55M25

Generated on Fri Feb 9 20:15:06 2018 by

LaTeXML
z_0 is a pole of f

z

0

is a pole of

f

Let

f

be an

analytic function

on a punctured

neighborhood

of

x

0

∈

𝐂

,
that is,

f

analytic on

{

z

∈

C

:

0

<

|

z

-

x

0

|

<

ε

}

for some

ε

>

0

and such that

lim

z

→

z

0

⁡

|

f

⁢

(

z

)

|

=

∞

.

We say then that

x

0

is a pole for

f

.

Title

z

0

is a pole of

f

Canonical

name

Z0IsAPoleOfF

Date of creation

2013-03-22 14:00:55

Last modified on

2013-03-22 14:00:55

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 30A99

Generated on Fri Feb 9 20:15:09 2018 by

LaTeXML
analytic

analytic

Let

U

be a domain in the

complex numbers

(resp.,

real numbers

). A

function

f

:

U

⟶

ℂ

(resp.,

f

:

U

⟶

ℝ

) is

analytic

(resp.,

real analytic

) if

f

has a

Taylor series

about each point

x

∈

U

that

converges

to the function

f

in an open neighborhood of

x

.

1

On Analyticity and Holomorphicity

A

complex function

is analytic if and only if it is

holomorphic

. Because of this equivalence, an

analytic function

in the

complex

case is often defined to be one that is holomorphic, instead of one having a Taylor series as above. Although the two definitions are

equivalent

, it is not an easy matter to prove their equivalence, and a reader who does not yet have this result available will have to pay attention as to which definition of analytic is being used.

Title

analytic

Canonical name

Analytic

Date of creation

2013-03-22 12:04:36

Last modified on

2013-03-22 12:04:36

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 30B10

Classification

msc 26A99

Synonym

real analytic

Synonym

real analytic function

Synonym

complex analytic

Synonym

complex analytic function

Synonym

analytic function

Related topic

Holomorphic

Related topic

TaylorSeries

Related topic

CauchyKowalewskiTheorem

Generated on Fri Feb 9 20:15:11 2018 by

LaTeXML
coefficients of Laurent series

coefficients of Laurent series

Suppose that

f

is

analytic

in the

annulus

{

z

∈

ℂ

⁢

⋮

⁢

R

1

<

|

z

-

a

|

<

R

2

}

,  where

R

1

may be 0 and

R

2

may be

∞

.  Then the coefficients of the

Laurent series

(

http://planetmath.org/LaurentSeries

)

∑

n

=

-

∞

∞

c

n

⁢

(

z

-

a

)

n

of

f

can be obtained from

c

n

=

1

2

⁢

π

⁢

i

∮

γ

f

⁢

(

t

)

(

t

-

a

)

n

+

1

d

t

(

n

=

0

,

±

1

,

±

2

,

…

)

,

(1)

where the path (

http://planetmath.org/ContourIntegral

)

γ

goes anticlockwise once around the point

z

=

a

within the annulus.  Especially, the

residue

of

f

in the point

a

is

c

-

1

=

1

2

⁢

π

⁢

i

⁢

∮

γ

f

⁢

(

t

)

⁢

𝑑

t

.

(2)

Remark.

Usually, the Laurent series of a

function

, i.e. the coefficients

c

n

, are not determined by using the

integral

formula (1), but directly from known series .  Often it is sufficient to know the value of

c

-

1

or the residue, which is used to compute integrals (see the

Cauchy residue theorem

—  cf. (2)).  There is also the usable

Rule.

In the case that the limit

lim

z

→

a

⁡

(

z

-

a

)

⁢

f

⁢

(

z

)

exists and has a non-zero value

r

, the point

z

=

a

is a pole of the 1 for the function

f

and

Res

⁡

(

f

;

a

)

=

r

.

Examples

1.

Let

f

⁢

(

z

)

:=

1

sin

⁡

z

,  and

a

=

0

.  Using the

Taylor series

of the

complex sine

we obtain

lim

z

→

0

⁡

z

⁢

1

sin

⁡

z

=

lim

z

→

0

⁡

1

1

-

z

2

3

!

+

-

…

=

1

,

whence

Res

⁡

(

1

sin

⁡

z

;

0

)

=

1

.  Thus we can write

∮

γ

d

⁢

z

sin

⁡

z

=

2

⁢

π

⁢

i

,

where the must be chosen such that it encloses only the pole

0

of

1

sin

⁡

z

.

2.

The Taylor series of the

complex exponential function

gives the Laurent series

e

1

z

≡

1

+

1

z

+

1

2

!

⁢

z

2

+

1

3

!

⁢

z

3

+

…

which shows that

Res

⁡

(

e

1

z

;

0

)

=

1

.

Title

coefficients of Laurent series

Canonical name

CoefficientsOfLaurentSeries

Date of creation

2013-03-22 15:19:22

Last modified on

2013-03-22 15:19:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Result

Classification

msc 30B10

Related topic

LaurentSeries

Related topic

TechniqueForComputingResidues

Related topic

UniquenessOfLaurentExpansion

Generated on Fri Feb 9 20:15:13 2018 by

LaTeXML
Euler relation

Euler relation

Euler’s relation

(also known as

Euler’s formula

) is considered the first between the fields of algebra and geometry, as it relates the

exponential function

to the trigonometric sine and

cosine functions

.

Euler’s relation states that

e

i

⁢

x

=

cos

⁡

x

+

i

⁢

sin

⁡

x

Start by noting that

i

k

=

{

1

if

⁢

k

≡

0

(

mod

4

)

i

if

⁢

k

≡

1

(

mod

4

)

-

1

if

⁢

k

≡

2

(

mod

4

)

-

i

if

⁢

k

≡

3

(

mod

4

)

Using the

Taylor series

expansions of

e

x

,

sin

⁡

x

and

cos

⁡

x

(see the entries on the

complex exponential function

and the

complex sine and cosine

), it follows that

e

i

⁢

x

=

∑

n

=

0

∞

i

n

⁢

x

n

n

!

=

∑

n

=

0

∞

(

x

4

⁢

n

(

4

⁢

n

)

!

+

i

⁢

x

4

⁢

n

+

1

(

4

⁢

n

+

1

)

!

-

x

4

⁢

n

+

2

(

4

⁢

n

+

2

)

!

-

i

⁢

x

4

⁢

n

+

3

(

4

⁢

n

+

3

)

!

)

Because the

series expansion

above is

absolutely convergent

for all

x

,
we can rearrange the terms of the series as

e

i

⁢

x

=

∑

n

=

0

∞

(

-

1

)

n

⁢

x

2

⁢

n

(

2

⁢

n

)

!

+

i

⁢

∑

n

=

0

∞

(

-

1

)

n

⁢

x

2

⁢

n

+

1

(

2

⁢

n

+

1

)

!

=

cos

⁡

x

+

i

⁢

sin

⁡

x

As a special case, we get the beautiful and well-known

identity

, often called

Euler’s identity

:

e

i

⁢

π

=

-

1

Title

Euler relation

Canonical name

EulerRelation

Date of creation

2013-03-22 11:57:05

Last modified on

2013-03-22 11:57:05

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

17

Author

rm50 (10146)

Entry type

Definition

Classification

msc 30B10

Synonym

Euler’s formula

Related topic

TaylorSeries

Related topic

DeMoivreIdentity

Related topic

ComplexSineAndCosine

Defines

Euler identity

Defines

Euler’s identity

Generated on Fri Feb 9 20:15:16 2018 by

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Fresnel integrals

Fresnel integrals

0.1

The functions

C

and

S

For any real value of the

argument

x

, the

Fresnel integrals

C

⁢

(

x

)

and

S

⁢

(

x

)

are defined as the

integrals

C

⁢

(

x

)

:=

∫

0

x

cos

⁡

t

2

⁢

d

⁢

t

,

S

⁢

(

x

)

:=

∫

0

x

sin

⁡

t

2

⁢

d

⁢

t

.

(1)

In optics, both of them express the .

Using the

Taylor series

expansions of cosine and sine (

http://planetmath.org/ComplexSineAndCosine

), we get easily the expansions of the

functions

(1):

C

(

z

)

=

z

1

-

z

5

5

⋅

2

!

+

z

9

9

⋅

4

!

-

z

13

13

⋅

6

!

+

-

…

S

(

z

)

=

z

3

3

⋅

1

!

-

z

7

7

⋅

3

!

+

z

11

11

⋅

5

!

-

z

15

15

⋅

7

!

+

-

…

These

converge

for all complex values

z

and thus define

entire transcendental functions

.

The

Fresnel integrals at infinity

have the finite value

lim

x

→

∞

⁡

C

⁢

(

x

)

=

lim

x

→

∞

⁡

S

⁢

(

x

)

=

2

⁢

π

4

.

0.2

Clothoid

The

parametric presentation

x

=

C

⁢

(

t

)

,

y

=

S

⁢

(

t

)

(2)

a curve called

clothoid

.  Since the equations (2) both define

odd functions

, the clothoid has symmetry about the origin.  The curve has the shape of a “

∼

”
(see this

http://www.wakkanet.fi/ pahio/A/A/clothoid.png

diagram).

The

arc length

of the clothoid from the origin to the point

(

C

⁢

(

t

)

,

S

⁢

(

t

)

)

is simply

∫

0

t

C

′

⁢

(

u

)

2

+

S

′

⁢

(

u

)

2

⁢

𝑑

u

=

∫

0

t

cos

2

⁡

(

u

2

)

+

sin

2

⁡

(

u

2

)

⁢

𝑑

u

=

∫

0

t

𝑑

u

=

t

.

Thus the of the whole curve (to the point

(

2

⁢

π

4

,

2

⁢

π

4

)

) is

infinite

.

The

curvature

(

http://planetmath.org/CurvaturePlaneCurve

) of the clothoid also is extremely ,

ϰ

=

2

⁢

t

,

i.e. proportional (

http://planetmath.org/Variation

) to the arc lenth; thus in the origin only the curvature is zero.

Conversely

, if the curvature of a plane curve varies proportionally to the arc length, the curve is a clothoid.

This

property

of the curvature of clothoid is utilised in way and railway construction, since the form of the clothoid is very efficient when a portion of way must be bent to a turn:  the zero curvature of the line can be continuously raised to the wished curvature.

Title

Fresnel integrals

Canonical

name

FresnelIntegrals

Date of creation

2014-07-11 21:15:59

Last modified on

2014-07-11 21:15:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

24

Author

pahio (2872)

Entry type

Definition

Classification

msc 30B10

Classification

msc 26A42

Classification

msc 33B20

Related topic

SineIntegral

Defines

clothoid

Generated on Fri Feb 9 20:15:22 2018 by

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identity theorem of power series

identity theorem of power series

If the radii of

convergence

(

http://planetmath.org/RadiusOfConvergence

) of the

power series

∑

n

=

0

∞

a

n

⁢

(

z

-

z

0

)

n

and

∑

n

=

0

∞

b

n

⁢

(

z

-

z

0

)

n

are positive and the sums of the series are equal in infinitely many points which have

z

0

as an

accumulation point

, then the both series are identical, i.e.

a

n

=

b

n

for each

n

=

0

,

1

,

2

,

…

Title

identity theorem of power series

Canonical name

IdentityTheoremOfPowerSeries

Date of creation

2013-03-22 16:47:08

Last modified on

2013-03-22 16:47:08

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30B10

Classification

msc 40A30

Related topic

IdentityTheoremOfHolomorphicFunctions

Related topic

TheoremsOnComplexFunctionSeries

Generated on Fri Feb 9 20:15:23 2018 by

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