aarjaneiro/planetmath huggingface dataset:
D’Alembertian

D’Alembertian

The D’Alembertian is the

equivalent

of the Laplacian in Minkowskian

geometry

. It is given by:

□

=

∇

2

-

1

c

2

⁢

∂

2

∂

⁡

t

2

Here we assume a Minkowskian metric of the form

(

+

,

+

,

+

,

-

)

as typically seen in special relativity. The

connection

between the Laplacian in

Euclidean space

and the D’Alembertian is clearer if we write both

operators

and their corresponding metric.

0.1

Laplacian

Metric:

⁢

d

⁢

s

2

=

d

⁢

x

2

+

d

⁢

y

2

+

d

⁢

z

2

Operator:

⁢

∇

2

=

∂

2

∂

⁡

x

2

+

∂

2

∂

⁡

y

2

+

∂

2

∂

⁡

z

2

0.2

D’Alembertian

Metric:

⁢

d

⁢

s

2

=

d

⁢

x

2

+

d

⁢

y

2

+

d

⁢

z

2

-

c

⁢

d

⁢

t

2

Operator:

⁢

□

=

∂

2

∂

⁡

x

2

+

∂

2

∂

⁡

y

2

+

∂

2

∂

⁡

z

2

-

1

c

2

⁢

∂

2

∂

⁡

t

2

In both cases we simply differentiate twice with respect to each coordinate in the metric. The D’Alembertian is hence a special case of the generalised Laplacian.

1

Connection with the wave equation

The wave equation is given by:

∇

2

⁡

u

=

1

c

2

⁢

∂

2

∂

⁡

t

2

⁢

u

Factorising in terms of operators, we obtain:

(

∇

2

-

1

c

2

⁢

∂

2

∂

⁡

t

2

)

⁢

u

=

0

or

□

⁢

u

=

0

Hence the frequent appearance of the D’Alembertian in special relativity and electromagnetic theory.

2

Alternative notation

The symbols

□

and

□

2

are both used for the D’Alembertian. Since it is unheard of to square the D’Alembertian, this is not as confusing as it may appear. The symbol for the Laplacian,

Δ

or

∇

2

, is often used when it is clear that a

Minkowski space

is being referred to.

3

Alternative definition

It is common to define Minkowski space to have the metric

(

-

,

+

,

+

,

+

)

, in which case the D’Alembertian is simply the

negative

of that defined above:

□

=

1

c

2

⁢

∂

2

∂

⁡

t

2

-

∇

2

Title

D’Alembertian

Canonical

name

DAlembertian

Date of creation

2013-03-22 17:55:18

Last modified on

2013-03-22 17:55:18

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

8

Author

invisiblerhino (19637)

Entry type

Definition

Classification

msc 31B15

Classification

msc 31B05

Classification

msc 26B12

Synonym

wave operator

Synonym

D’Alembert operator

Related topic

Laplacian

Generated on Fri Feb 9 20:30:30 2018 by

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Neumann problem

Neumann problem

Suppose

Ω

is a region of

ℝ

n

and

∂

⁡

Ω

is the boundary of

Ω

.
Further suppose

f

is a

function

f

:

∂

⁡

Ω

→

ℂ

, and suppose

∂

∂

⁡

n

corresponds to taking a

derivative

in a direction normal to the boundary

∂

⁡

Ω

at any point. Then the

Neumann problem

is to find a function

ϕ

:

Ω

∪

∂

⁡

Ω

→

ℂ

such that

∂

⁡

ϕ

∂

⁡

n

=

f

,

on

∂

⁡

Ω

,

∇

2

⁡

ϕ

=

0

,

in

Ω

.

Here

∇

2

represents the Laplacian operator and the second condition is that

ϕ

be a

harmonic function

on

Ω

. The condition for the

existence

of a solution

ϕ

of the Neumann problem is that integral of the normal derivative of the function

ϕ

, calculated over the entire boundary

∂

⁡

Ω

, vanish. This follows from the

identic equation

∫

∂

⁡

Ω

∂

⁡

ϕ

∂

⁡

n

⁢

𝑑

σ

=

∫

Ω

∇

⋅

(

∇

⁡

ϕ

)

⁢

𝑑

τ

=

∫

Ω

∇

2

⁡

ϕ

⁢

d

⁢

τ

and from the fact that

∇

2

⁡

ϕ

=

0

.

Title

Neumann problem

Canonical name

NeumannProblem

Date of creation

2013-03-22 15:19:59

Last modified on

2013-03-22 15:19:59

Owner

dczammit (9747)

Last modified by

dczammit (9747)

Numerical id

10

Author

dczammit (9747)

Entry type

Definition

Classification

msc 31B15

Classification

msc 31B05

Classification

msc 31A05

Related topic

HarmonicFunction

Generated on Fri Feb 9 20:30:32 2018 by

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harmonic function

harmonic function

A twice-differentiable real or complex-valued function

f

:

U

→

ℝ

or

f

:

U

→

ℂ

, where

U

⊆

ℝ

n

is some , is called

harmonic

if its Laplacian vanishes on

U

, i.e. if

Δ

⁢

f

≡

0

.

Any

harmonic function

f

:

ℝ

n

→

ℝ

or

f

:

ℝ

n

→

ℂ

satisfies Liouville’s

theorem

. Indeed, a

holomorphic function

is

harmonic, and a real harmonic function

f

:

U

→

ℝ

, where

U

⊆

ℝ

2

, is locally the

real part

of a holomorphic function. In fact, it is enough that a harmonic function

f

be below (or above) to conclude that it is .

Title

harmonic function

Canonical

name

HarmonicFunction

Date of creation

2013-03-22 12:43:46

Last modified on

2013-03-22 12:43:46

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 31C05

Classification

msc 31B05

Classification

msc 31A05

Classification

msc 30F15

Related topic

RadosTheorem

Related topic

SubharmonicAndSuperharmonicFunctions

Related topic

DirichletProblem

Related topic

NeumannProblem

Generated on Fri Feb 9 20:30:33 2018 by

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pluriharmonic function

pluriharmonic function

Definition

.

Let

f

:

G

⊂

ℂ

n

→

ℂ

be a

C

2

(twice

continuously differentiable

)

function

.

f

is called

pluriharmonic

if for every

complex line

{

a

+

b

⁢

z

∣

z

∈

ℂ

}

the function

z

↦

f

⁢

(

a

+

b

⁢

z

)

is

harmonic

on the set

{

z

∈

ℂ

∣

a

+

b

⁢

z

∈

G

}

.

Note that every pluriharmonic function is a

harmonic function

, but not the other way around. Further it can be shown that for

holomorphic functions

of several complex variables the real (and the imaginary) parts are locally pluriharmonic functions. Do note however that just because a function is harmonic in each variable separately does not imply that it is pluriharmonic.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

pluriharmonic function

Canonical name

PluriharmonicFunction

Date of creation

2013-03-22 14:29:01

Last modified on

2013-03-22 14:29:01

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 31C05

Classification

msc 32A50

Classification

msc 31C10

Synonym

pluriharmonic

Generated on Fri Feb 9 20:30:35 2018 by

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polar set

polar set

Definition

.

Let

G

⊂

ℝ

n

and let

f

:

G

→

ℝ

∪

{

-

∞

}

be a subharmonic
function which is not identically

-

∞

.
The set

𝒫

:=

{

x

∈

G

∣

f

⁢

(

x

)

=

-

∞

}

is
called a

polar set

.

Proposition

.

Let

G

and

P

be as above and suppose that

g

is a

continuous

subharmonic function on

G

∖

P

. Then

g

is subharmonic
on the entire set

G

.

The requirement that

g

is continuous cannot be relaxed.

Proposition

.

Let

G

and

P

be as above. Then the

Lebesgue measure

of

P

is 0.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

polar set

Canonical

name

PolarSet

Date of creation

2013-03-22 14:29:13

Last modified on

2013-03-22 14:29:13

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 31C05

Classification

msc 31B05

Classification

msc 31A05

Generated on Fri Feb 9 20:30:37 2018 by

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plurisubharmonic function

plurisubharmonic function

Definition

.

Let

f

:

G

⊂

ℂ

n

→

ℝ

be an

upper semi-continuous

function

.

f

is called

plurisubharmonic

if for every

complex line

{

a

+

b

⁢

z

∣

z

∈

ℂ

}

the function

z

↦

f

⁢

(

a

+

b

⁢

z

)

is a

subharmonic function

on the set

{

z

∈

ℂ

∣

a

+

b

⁢

z

∈

G

}

.

Similarly, we could also define a

plurisuperharmonic

function just like
we have a superharmonic function, but again it just means that

-

f

is
plurisubharmonic, and so this extra is not very useful.

Definition

.

A

continuous

plurisubharmonic function

is said to be a

pseudoconvex function

.

Note that since plurisubharmonic is a long word, many authors abbreviate
with

psh

,

plsh

, or

plush

.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

plurisubharmonic function

Canonical name

PlurisubharmonicFunction

Date of creation

2013-03-22 14:29:09

Last modified on

2013-03-22 14:29:09

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Definition

Classification

msc 31C10

Classification

msc 32U05

Synonym

plurisubharmonic

Defines

plurisuperharmonic function

Defines

pseudoconvex function

Generated on Fri Feb 9 20:30:41 2018 by

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complex Hessian matrix

complex Hessian matrix

Suppose that

f

:

ℂ

n

→

ℂ

be twice

differentiable

and let

∂

∂

⁡

z

k

:=

1

2

⁢

(

∂

∂

⁡

x

k

-

i

⁢

∂

∂

⁡

y

k

)

and

∂

∂

⁡

z

¯

k

:=

1

2

⁢

(

∂

∂

⁡

x

k

+

i

⁢

∂

∂

⁡

y

k

)

.

Then the is the matrix

[

∂

2

⁡

f

∂

⁡

z

1

⁢

∂

⁡

z

¯

1

∂

2

⁡

f

∂

⁡

z

1

⁢

∂

⁡

z

¯

2

…

∂

2

⁡

f

∂

⁡

z

1

⁢

∂

⁡

z

¯

n

∂

2

⁡

f

∂

⁡

z

2

⁢

∂

⁡

z

¯

1

∂

2

⁡

f

∂

⁡

z

2

⁢

∂

⁡

z

¯

2

…

∂

2

⁡

f

∂

⁡

z

2

⁢

∂

⁡

z

¯

n

⋮

⋮

⋱

⋮

∂

2

⁡

f

∂

⁡

z

n

⁢

∂

⁡

z

¯

1

∂

2

⁡

f

∂

⁡

z

n

⁢

∂

⁡

z

¯

2

…

∂

2

⁡

f

∂

⁡

z

n

⁢

∂

⁡

z

¯

n

]

.

When applied to

tangent vectors

of the

zero set

of

f

,
it is called the

Levi form

and used to define a Levi
pseudoconvex point of a boundary of a domain. Note that the
matrix is not the same as the (real)

Hessian

. A twice continuously
differentiable real valued
function with a

positive semidefinite

real Hessian matrix at every point is convex, but a function with

positive semidefinite matrix

at every point is

plurisubharmonic

(since it’s

continuous

it’s also called a pseudoconvex function).

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

complex Hessian matrix

Canonical name

ComplexHessianMatrix

Date of creation

2013-03-22 14:31:16

Last modified on

2013-03-22 14:31:16

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32-00

Related topic

HessianMatrix

Generated on Fri Feb 9 20:31:04 2018 by

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complex line

complex line

Definition

.

Let

a

,

b

∈

ℂ

n

. The set

ℓ

:=

{

a

+

b

⁢

z

∣

z

∈

ℂ

}

is called the

complex line

.

A complex line is a

holomorphic

complex affine imbedding of

ℂ

into

ℂ

n

so that if

f

is holomorphic, then

z

↦

f

⁢

(

a

+

b

⁢

z

)

is also holomorphic. That is the complex structures of

ℓ

and

ℂ

n

are

compatible

. Hence not every two dimensional real affine space is a complex line.

Definition

.

Let

a

,

b

1

,

…

,

b

k

∈

ℂ

n

such that

b

1

,

…

,

b

k

are

linearly independent

over

ℂ

,
then. The set

{

a

+

∑

j

=

1

k

b

k

⁢

z

k

∣

z

1

,

…

,

z

k

∈

ℂ

}

is called the

complex affine space

.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

complex line

Canonical name

ComplexLine

Date of creation

2013-03-22 14:29:05

Last modified on

2013-03-22 14:29:05

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32-00

Related topic

AffineTransformation

Defines

complex affine space

Generated on Fri Feb 9 20:31:06 2018 by

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modulus of complex number

modulus of complex number

Definition

Let

z

be a

complex number

, and let

z

¯

be the

complex conjugate

of

z

.
Then the

modulus

, or

absolute value

, of

z

is defined as

|

z

|

:=

z

⁢

z

¯

.

There is also the notation

mod

z

for the modulus of

z

.

If we write

z

in

polar form

as

z

=

r

⁢

e

i

⁢

ϕ

with

r

≥

0

,

ϕ

∈

[

0

,

2

⁢

π

)

,
then

|

z

|

=

r

. It follows that the modulus is a positive real number or zero.
Alternatively, if

a

is the

real part

of

z

, and

b

the

imaginary part

, then

|

z

|

=

a

2

+

b

2

,

(1)

which is simply the

Euclidean norm

of the point

(

a

,

b

)

∈

ℝ

2

.
It follows that the modulus satisfies the

triangle inequality

|

z

1

+

z

2

|

≤

|

z

1

|

+

|

z

2

|

,

also

|

ℜ

⁡

z

|

≤

|

z

|

,

|

ℑ

⁡

z

|

≤

|

z

|

,

|

z

|

≤

|

ℜ

⁡

z

|

+

|

ℑ

⁡

z

|

.

Modulus is :

|

z

1

⁢

z

2

|

=

|

z

1

|

⋅

|

z

2

|

,

|

z

1

z

2

|

=

|

z

1

|

|

z

2

|

Since

ℝ

⊂

ℂ

, the definition of modulus includes the

real numbers

. Explicitly, if we write

x

∈

ℝ

in polar form,

x

=

r

⁢

e

i

⁢

ϕ

,

r

>

0

,

ϕ

∈

[

0

,

2

⁢

π

)

,  then

ϕ

=

0

or

ϕ

=

π

, so

e

i

⁢

ϕ

=

±

1

. Thus,

|

x

|

=

x

2

=

{

x

x

>

0

0

x

=

0

-

x

x

<

0

.

Title

modulus of complex number

Canonical name

ModulusOfComplexNumber

Date of creation

2013-03-22 13:36:39

Last modified on

2013-03-22 13:36:39

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

17

Author

matte (1858)

Entry type

Definition

Classification

msc 32-00

Classification

msc 30-00

Classification

msc 12D99

Synonym

complex modulus

Synonym

modulus

Synonym

absolute value of complex number

Synonym

absolute value

Synonym

modulus of a complex number

Related topic

AbsoluteValue

Related topic

Subadditive

Related topic

SignumFunction

Related topic

ComplexConjugate

Related topic

PotentialOfHollowBall

Related topic

ConvergenceOfRiemannZetaSeries

Related topic

RealPartSeriesAndImaginaryPartSeries

Related topic

ArgumentOfProductAndSum

Related topic

ArgumentOfProductAndQuotient

Related topic

EqualityOfComplexNumbers

Generated on Fri Feb 9 20:31:08 2018 by

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complex logarithm

complex logarithm

The

z

is defined as every

complex number

w

which satisfies the equation

e

w

=

z

.

(1)

This is is denoted by

log

⁡

z

:=

w

.

The solution of (1) is obtained by using the form

e

w

=

r

⁢

e

i

⁢

φ

, where

r

=

|

z

|

and

φ

=

arg

⁡

z

;  the result is

w

=

log

⁡

z

=

ln

⁡

|

z

|

+

i

⁢

arg

⁡

z

.

Here, the

ln

⁡

|

z

|

means the usual Napierian or

natural logarithm

(

http://planetmath.org/NaturalLogarithm2

) (‘

logarithmus naturalis

’) of the

real number

|

z

|

.  If we fix the phase angle

φ

of

|

z

|

so that

0

≦

φ

<

2

⁢

π

, we can write

log

z

=

ln

r

+

i

φ

+

n

⋅

2

π

i

(

n

=

0

,

±

1

,

±

2

,

…

)

.

The complex logarithm

log

⁡

z

is defined for all

z

≠

0

and it is infinitely multivalued

-

e.g.

log

⁡

(

-

1

)

=

(

2

⁢

n

+

1

)

⁢

π

⁢

i

where

n

is an arbitrary integer.  The values with

n

=

0

are called the of the ; if

z

is real, the value of

log

⁡

z

coincides with

ln

⁡

z

.

Title

complex logarithm

Canonical name

ComplexLogarithm

Date of creation

2013-03-22 14:43:11

Last modified on

2013-03-22 14:43:11

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 32A05

Classification

msc 30D20

Synonym

natural logarithm

Related topic

Logarithm

Related topic

NaturalLogarithm2

Related topic

ValuesOfComplexCosine

Related topic

EqualityOfComplexNumbers

Related topic

SomeValuesCharacterisingI

Related topic

UsingResidueTheoremNearBranchPoint

Generated on Fri Feb 9 20:31:11 2018 by

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derivative of exponential function

derivative of exponential function

In this entry, we shall compute the

derivative

of the

exponential
function

from its

definition

as a limit of powers.

Theorem 1

.

If

0

≤

x

<

1

, then

1

+

x

≤

exp

⁡

x

≤

1

1

-

x

Proof.

By the

inequalities for differences of powers

, we have

x

≤

(

1

+

x

n

)

n

-

1

≤

x

1

-

(

n

-

1

n

)

⁢

x

.

Since

n

-

1

<

n

, and

x

>

0

, we have

0

<

(

n

-

1

/

n

)

⁢

x

<

x

. Because

x

<

1

, this implies

1

-

(

n

-

1

/

n

)

⁢

x

>

1

-

x

, so

x

1

-

(

n

-

1

n

)

⁢

x

<

x

1

-

x

.

Hence

1

+

x

≤

(

1

+

x

n

)

n

≤

1

1

-

x

.

Taking the limit as

n

→

∞

, we obtain our result.
∎

Theorem 2

.

lim

x

→

0

⁡

exp

⁡

(

x

)

-

1

x

=

1

Proof.

Assume

0

<

x

<

1

. By our bound, we have

1

≤

exp

⁡

(

x

)

-

1

x

≤

1

1

-

x

.

Suppose that

-

1

<

x

<

0

. Then, since

exp

⁡

(

x

)

=

1

/

exp

⁡

(

-

x

)

, we have

exp

⁡

(

x

)

-

1

x

=

1

exp

⁡

(

-

x

)

⋅

1

-

exp

⁡

(

-

x

)

x

.

From the

inequality

above, we have

1

≤

1

-

exp

⁡

(

-

x

)

x

≤

1

1

+

x

.

Hence

1

exp

⁡

(

-

x

)

≤

exp

⁡

(

x

)

-

1

x

≤

1

(

1

+

x

)

⁢

exp

⁡

(

-

x

)

.

By

theorem

1, we have

1

-

x

≤

exp

⁡

(

-

x

)

≤

1

/

(

1

+

x

)

, so

1

+

x

≤

1

-

exp

⁡

(

-

x

)

x

≤

1

(

1

+

x

)

⁢

(

1

-

x

)

=

1

1

-

x

2

.

By the

squeeze rule

, we conclude that

lim

x

→

0

⁡

1

-

exp

⁡

(

-

x

)

x

=

1

whether we approach the limit from the left or the right.
∎

Theorem 3

.

d

d

⁢

x

⁢

exp

⁡

(

x

)

=

exp

⁡

(

x

)

Proof.

By definition,

d

d

⁢

x

⁢

exp

⁡

(

x

)

=

lim

y

→

x

⁡

exp

⁡

(

y

)

-

exp

⁡

(

x

)

y

-

x

.

By the

addition theorem

for the

exponential

, we have

exp

⁡

(

y

)

-

exp

⁡

(

x

)

y

-

x

=

exp

⁡

(

x

)

⋅

exp

⁡

(

y

-

x

)

-

1

y

-

x

,

so

lim

y

→

x

⁡

exp

⁡

(

y

)

-

exp

⁡

(

x

)

y

-

x

=

exp

⁡

(

x

)

⁢

lim

y

→

x

⁡

exp

⁡

(

y

-

x

)

-

1

y

-

x

=

exp

⁡

(

x

)

⁢

lim

y

→

0

⁡

exp

⁡

y

-

1

y

.

By theorem 2, the limit on the right-hand side equals

1

, so we have

lim

y

→

x

⁡

exp

⁡

(

y

)

-

exp

⁡

(

x

)

y

-

x

=

exp

⁡

(

x

)

.

∎

Title

derivative of exponential function

Canonical

name

DerivativeOfExponentialFunction

Date of creation

2013-03-22 17:01:39

Last modified on

2013-03-22 17:01:39

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

15

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 32A05

Related topic

ExponentialFunction

Related topic

ComplexExponentialFunction

Related topic

DerivativeOfTheNaturalLogarithmFunction

Generated on Fri Feb 9 20:31:14 2018 by

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exponential function never vanishes

exponential function never vanishes

In the entry

exponential function

(

http://planetmath.org/ExponentialFunction

) one defines for real variable

x

the real exponential function

exp

⁡

x

, i.e.

e

x

, as the sum of power series:

e

x

=

∑

k

=

0

∞

x

k

k

!

The series form implies immediately that the real exponential
function attains only

positive

values when

x

≧

0

.  Also
for

-

1

≦

x

<

0

the positiveness is easy to see by
grouping the series terms pairwise.

In to study the sign of

e

x

for
arbitrary real

x

, we may multiply the series of

e

x

and

e

-

x

using Abel’s

multiplication

rule for series (

http://planetmath.org/AbelsMultiplicationRuleForSeries

).  We obtain

e

x

⁢

e

-

x

=

∑

n

=

0

∞

x

n

n

!

⁢

∑

k

=

0

∞

(

-

1

)

k

⁢

x

k

k

!

=

∑

n

=

0

∞

∑

j

=

0

n

(

-

1

)

j

⁢

x

n

j

!

⁢

(

n

-

j

)

!

=

∑

n

=

0

∞

x

n

n

!

⁢

∑

j

=

0

n

(

n

j

)

⁢

(

-

1

)

j

=

∑

n

=

0

∞

x

n

n

!

⋅

0

n

.

The last sum equals 1.  So, if

-

x

>

0

,  then

e

-

x

>

0

,  whence

e

x

must be positive.

Let us now consider arbitrary

complex

value

z

=

x

+

i

⁢

y

where

x

and

y

are real.  Using the

addition formula

of

complex exponential function

and the

Euler relation

, we can write

e

z

=

e

x

+

i

⁢

y

=

e

x

⁢

e

i

⁢

y

=

e

x

⁢

(

cos

⁡

y

+

i

⁢

sin

⁡

y

)

.

From this we see that the

absolute value

of

e

z

is

e

x

, which we above have proved to be positive.  Accordingly, we may write the

Theorem

.

The complex exponential function never vanishes.

Title

exponential function never vanishes

Canonical name

ExponentialFunctionNeverVanishes

Date of creation

2014-11-22 21:23:32

Last modified on

2014-11-22 21:23:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 32A05

Classification

msc 30D20

Synonym

real exponential function is positive

Related topic

ExponentialFunction

Related topic

ExponentialFunctionDefinedAsLimitOfPowers

Related topic

PeriodicityOfExponentialFunction

Generated on Fri Feb 9 20:31:29 2018 by

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generating function for the reciprocal alternating central binomial coefficients

generating function for the reciprocal alternating central binomial coefficients

It is also not very well known this

relation

:

4

⁢

(

x

+

4

-

x

⁢

arcsinh

⁢

(

x

2

)

)

(

x

+

4

)

3

=

1

-

x

2

+

x

2

6

-

x

3

20

+

x

4

70

-

x

5

252

+

x

6

924

-

…

where one clearly appreciate that the

function

on LHS generates the

sequence

(

-

1

)

n

⁢

(

2

⁢

n

n

)

-

1

.

To obtain the relation above one should use some kind of software because for the function is “terrible” to calculate

derivatives

of any order. It is a little challenge to give a

recursive

formula

that gives the

inverses

of these alternating

central

binomial numbers, when evaluated at

x

=

0

at those derivatives.

Title

generating function

for the reciprocal alternating

central binomial coefficients

Canonical name

GeneratingFunctionForTheReciprocalAlternatingCentralBinomialCoefficients

Date of creation

2013-03-22 18:58:12

Last modified on

2013-03-22 18:58:12

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

14

Author

juanman (12619)

Entry type

Example

Classification

msc 32A05

Classification

msc 11B65

Classification

msc 05A19

Classification

msc 05A15

Classification

msc 05A10

Generated on Fri Feb 9 20:31:31 2018 by

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limit of {a^x-1}/x as x approaches 0

limit of

a

x

-

1

x

as

x

approaches 0

Corollary

.

For

a

>

0

, we have

lim

x

→

0

⁡

a

x

-

1

x

=

ln

⁡

a

.

Proof.

Recall that

a

x

=

e

x

⁢

ln

⁡

a

. Thus,

lim

x

→

0

⁡

a

x

-

1

x

=

lim

x

→

0

⁡

e

x

⁢

ln

⁡

a

-

1

x

=

lim

x

→

0

⁡

(

e

x

⁢

ln

⁡

a

-

1

)

⁢

ln

⁡

a

x

⁢

ln

⁡

a

=

(

ln

⁡

a

)

⁢

lim

x

→

0

⁡

e

x

⁢

ln

⁡

a

-

1

x

⁢

ln

⁡

a

.

Let

t

=

x

⁢

ln

⁡

a

. Then

t

→

0

as

x

→

0

. Therefore,

lim

x

→

0

⁡

a

x

-

1

x

=

(

ln

⁡

a

)

⁢

lim

t

→

0

⁡

e

t

-

1

t

=

(

ln

⁡

a

)

⁢

1

=

ln

⁡

a

.

∎

The formula from the

corollary

is useful for proving that

d

d

⁢

x

⁢

a

x

=

a

x

⁢

ln

⁡

a

. On the other hand, once this fact is known, the corollary is easily proven via l’Hôpital’s rule (

http://planetmath.org/LHpitalsRule

):

lim

x

→

0

⁡

a

x

-

1

x

=

lim

x

→

0

⁡

a

x

⁢

ln

⁡

a

1

=

a

0

⁢

ln

⁡

a

=

ln

⁡

a

.

Title

limit of

a

x

-

1

x

as

x

approaches 0

Canonical

name

LimitOfdisplaystylefracax1xAsXApproaches0

Date of creation

2013-03-22 17:40:21

Last modified on

2013-03-22 17:40:21

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

5

Author

Wkbj79 (1863)

Entry type

Corollary

Classification

msc 32A05

Generated on Fri Feb 9 20:31:34 2018 by

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