aarjaneiro/planetmath huggingface dataset:
addition formulas

addition formulas

The

addition formula

of a real (

http://planetmath.org/RealFunction

) or

complex function

shows how the value of the

function

at a sum-formed variable can be expressed with the values of this function and perhaps of another function at the addends.

Examples

1.

Addition formula of an

additive function

f

,

f

⁢

(

x

+

y

)

=

f

⁢

(

x

)

+

f

⁢

(

y

)

2.

Addition formula of the

natural power function

, i.e. the

binomial theorem

,

(

x

+

y

)

n

=

∑

ν

=

0

n

(

n

ν

)

x

ν

y

n

-

ν

(

n

=

0

,

1

,

2

,

…

)

3.

Addition formula of the

exponential function

(

http://planetmath.org/ComplexExponentialFunction

),

e

x

+

y

=

e

x

⁢

e

y

4.

Addition formulae

of the

trigonometric functions

(

http://planetmath.org/DefinitionsInTrigonometry

), e.g.

cos

⁡

(

x

+

y

)

=

cos

⁡

x

⁢

cos

⁡

y

-

sin

⁡

x

⁢

sin

⁡

y

,

1

1

The addition formula of cosine is sometimes called “the mother of all formulae”.

⁢

tan

⁡

(

x

+

y

)

=

tan

⁡

x

+

tan

⁡

y

1

-

tan

⁡

x

⁢

tan

⁡

y

5.

Addition formulae of the

hyperbolic functions

, e.g.

sinh

⁡

(

x

+

y

)

=

sinh

⁡

x

⁢

cosh

⁡

y

+

cosh

⁡

x

⁢

sinh

⁡

y

6.

Addition formula of the

Bessel function

,

J

n

(

x

+

y

)

=

∑

ν

=

-

∞

∞

J

ν

(

x

)

J

n

-

ν

(

y

)

(

n

=

0

,

±

1

,

±

2

,

…

)

The five first of those are instances of ; e.g.

cosh

⁡

x

and

sinh

⁡

x

are tied together by the

algebraic

connection (

http://planetmath.org/UnitHyperbola

)

cosh

2

⁡

x

-

sinh

2

⁡

x

=

1

.

One may also speak of the

subtraction formulae

of functions — one example would be

e

x

-

y

=

e

x

e

y

.

Title

addition formulas

Canonical name

AdditionFormulas

Date of creation

2013-03-22 19:35:28

Last modified on

2013-03-22 19:35:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 30D05

Classification

msc 30A99

Classification

msc 26A99

Related topic

ExampleOnSolvingAFunctionalEquation

Related topic

ProofOfAdditionFormulaOfExp

Related topic

AdditionFormulasForSineAndCosine

Related topic

AdditionFormulaForTangent

Related topic

AdditionAndSubtractionFormulasForHyperbolicFunctions

Defines

addition formula

Defines

subtraction formula

Generated on Fri Feb 9 20:17:25 2018 by

LaTeXML
Plemelj formulas

Plemelj formulas

Let

ψ

⁢

(

ζ

)

be a

density function

of a complex variable satisfying the Hölder condition (the

Lipschitz condition

of order

α

)

1

1

A

function

f

⁢

(

ζ

)

satisfies

the Hölder condition on a smooth curve

C

if for every

ζ

1

,

ζ

2

∈

C

|

f

⁢

(

ζ

2

)

-

f

⁢

(

ζ

1

)

|

≤

M

⁢

|

ζ

2

-

ζ

1

|

α

,

M

>

0

,

0

<

α

≤

1

. It is clear that the Hölder condition is a weaker

restriction

than a

bounded

derivative

for

f

⁢

(

ζ

)

.

on a smooth closed

contour

C

in the

integral

Ψ

⁢

(

z

)

=

1

2

⁢

π

⁢

i

⁢

∫

C

ψ

⁢

(

ζ

)

ζ

-

z

⁢

𝑑

ζ

,

(1)

then the limits

Ψ

+

⁢

(

t

)

and

Ψ

-

⁢

(

t

)

as

z

approaches an arbitrary point

t

on

C

from the interior and the exterior of

C

, respectively, are

{

Ψ

+

⁢

(

t

)

≡

1

2

⁢

ψ

⁢

(

t

)

+

1

2

⁢

π

⁢

i

⁢

∫

C

ψ

⁢

(

ζ

)

ζ

-

t

⁢

𝑑

ζ

,

Ψ

-

⁢

(

t

)

≡

-

1

2

⁢

ψ

⁢

(

t

)

+

1

2

⁢

π

⁢

i

⁢

∫

C

ψ

⁢

(

ζ

)

ζ

-

t

⁢

𝑑

ζ

.

(2)

These are the Plemelj

[

1

]

formulas

2

2

cf.

[

2

]

, where restrictions that Plemelj made, were relaxed.

and the improper integrals in (2) must be interpreted as Cauchy’s

principal values

.

References

1

J. Plemelj,

Monatshefte für Mathematik und Physik,

vol. 19, pp. 205- 210, 1908.

2

N. I. Muskhelishvili,

Singular Integral Equations,

Groningen: Noordhoff (based on the second Russian edition published in 1946), 1953.

Title

Plemelj formulas

Canonical name

PlemeljFormulas

Date of creation

2013-03-22 16:02:02

Last modified on

2013-03-22 16:02:02

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

5

Author

perucho (2192)

Entry type

Definition

Classification

msc 30D10

Generated on Fri Feb 9 20:17:39 2018 by

LaTeXML
complex exponential function

complex exponential function

The

complex exponential function

exp

:

ℂ

→

ℂ

may be defined in many equivalent ways:  Let

z

=

x

+

i

⁢

y

where

x

,

y

∈

ℝ

.

•

exp

⁡

z

:=

e

x

⁢

(

cos

⁡

y

+

i

⁢

sin

⁡

y

)

•

exp

⁡

z

:=

lim

n

→

∞

⁡

(

1

+

z

n

)

n

•

exp

⁡

z

:=

∑

n

=

0

∞

z

n

n

!

The complex exponential function is usually denoted in power form:

e

z

:=

exp

⁡

z

,

where

e

is the Napier’s constant.  It also coincincides with the real

exponential function

when

z

is real (choose

y

=

0

).  It has all the

properties

of power, e.g.

e

-

z

=

1

e

z

;  these are consequences of the

addition formula

e

z

1

+

z

2

=

e

z

1

⁢

e

z

2

of the complex exponential function.

The

function

gets all

complex

values except 0 and is

periodic

(

http://planetmath.org/PeriodicityOfExponentialFunction

) having the (the with least non-zero

modulus

)

2

⁢

π

⁢

i

.  The

exp

is

holomorphic

, its

derivative

d

d

⁢

z

⁢

e

z

=

e

z

,

which is obtained from the series form via

termwise differentiation

, is similar as in

ℝ

.

So we have a fourth way to define

•

exp

⁡

z

:=

w

⁢

(

z

)

with

w

the solution of the

differential equation

d

⁢

w

d

⁢

z

=

w

under the

initial condition

w

⁢

(

0

)

=

1

.

Some formulae:

|

e

z

|

=

e

x

,

arg

e

z

=

y

+

2

n

π

(

n

=

0

,

±

1

,

±

2

,

…

)

,

Re

⁢

(

e

z

)

=

e

x

⁢

cos

⁡

y

,

Im

⁢

(

e

z

)

=

e

x

⁢

sin

⁡

y

Title

complex exponential function

Canonical

name

ComplexExponentialFunction

Date of creation

2013-03-22 14:43:08

Last modified on

2013-03-22 14:43:08

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

22

Author

pahio (2872)

Entry type

Definition

Classification

msc 30D20

Classification

msc 32A05

Related topic

ExponentialFunctionDefinedAsLimitOfPowers

Related topic

ExponentialFunction

Related topic

ComplexSineAndCosine

Related topic

ProofOfEquivalenceOfFormulasForExp

Related topic

DerivativeOfExponentialFunction

Related topic

ConvergenceOfRiemannZetaSeries

Defines

exponential function

Defines

prime period

Generated on Fri Feb 9 20:17:54 2018 by

LaTeXML
entire function

entire function

An

entire function

is a

function

f

:

ℂ

⟶

ℂ

which is

holomorphic

everywhere on the

complex

domain

ℂ

.

For example, a

polynomial

is holomorphic everywhere, as is the

exponential function

. The function

z

↦

1

/

z

is not holomorphic at zero, so it is not entire; it is

meromorphic

.

Title

entire function

Canonical

name

EntireFunction

Date of creation

2013-03-22 12:04:39

Last modified on

2013-03-22 12:04:39

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 30D20

Synonym

entire

Generated on Fri Feb 9 20:17:56 2018 by

LaTeXML
Gauss–Lucas theorem

Gauss–Lucas theorem

Theorem

.

If a convex polygon of the

complex plane

contains all zeros (

http://planetmath.org/ZeroOfAFunction

) of a

polynomial

f

⁢

(

z

)

, then it contains also all zeros of the derivative (

http://planetmath.org/ComplexDerivative

)

f

′

⁢

(

z

)

.

Proof.

Due to the

fundamental theorem of algebra

, the polynomial

f

⁢

(

z

)

can be written in the form

f

⁢

(

z

)

=

a

0

⁢

(

z

-

z

1

)

⁢

(

z

-

z

2

)

⁢

⋯

⁢

(

z

-

z

n

)

(1)

where

a

0

is the leading coefficient and

z

1

,

z

2

, …,

z

n

are the zeros of

f

⁢

(

z

)

(some of these may coincide).  When the derivative

f

′

⁢

(

z

)

=

∑

μ

=

1

n

∏

ν

≠

μ

(

z

-

z

ν

)

is divided by (1), we have the

identic equation

f

′

⁢

(

z

)

f

⁢

(

z

)

=

1

z

-

z

1

+

1

z

-

z

2

+

…

+

1

z

-

z

n

,

and therefore

f

′

⁢

(

z

)

=

f

⁢

(

z

)

⁢

(

1

z

-

z

1

+

1

z

-

z

2

+

…

+

1

z

-

z

n

)

.

(2)

Since

f

⁢

(

z

)

has no zeros outside the polygon, we have, according to (2), to show only that the same concerns the second factor of the right hand side of (2).

Let

z

be an arbitrary point outside the polygon.  Because of its convexity, there is a line

l

through

z

such that the polygon is completely on the other side of

l

.  Thus all directed line segments from

z

to the points

z

1

,

z

2

,

…

,

z

n

lie on the same side of the line

l

.  The direction angles of those

segments

, being the arguments of the

complex numbers

z

1

-

z

,

z

2

-

z

,

…

,

z

n

-

z

,  are between the values

α

and

α

+

π

, where

α

is one of the two angles which

l

forms with the

positive real axis

.

The arguments of any non-zero complex number and its

inverse number

are always opposite numbers of each other.  Hence the arguments of

1

z

-

z

1

,

1

z

-

z

2

,

…

,

1

z

-

z

n

(3)

are between

-

α

and

-

α

-

π

.  Think the line

m

through the origin with the direction angle

-

α

and the position vectors of the numbers (3).  These vectors are all directed on the same side of

m

, and similarly their sum vector, which thus is distinct from zero.  The opposite vector of this sum vector is equal to the second factor in (2), and accordingly distinct from zero.

Note.

The theorem can be strengthened to the following form:  The zeros of the derivative of any non-zero polynomial

f

⁢

(

z

)

are contained in the convex hull of the set of the zeros of

f

⁢

(

z

)

.

Title

Gauss–

Lucas theorem

Canonical

name

GaussLucasTheorem

Date of creation

2013-03-22 18:19:42

Last modified on

2013-03-22 18:19:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30D20

Synonym

Gauss-Lucas theorem

Synonym

Lucas’ theorem

Synonym

Lucas’s

root theorem

Synonym

zeros of polynomial derivative

Related topic

RollesTheorem

Related topic

ProductRule

Related topic

LogarithmicDerivative

Related topic

SlopeAngle

Related topic

FundamentalTheoremOfAlgebra

Generated on Fri Feb 9 20:17:58 2018 by

LaTeXML
holomorphic

holomorphic

Let

U

⊂

ℂ

be a domain in the

complex numbers

. A

function

f

:

U

⟶

ℂ

is

holomorphic

if

f

has a

complex derivative

at every point

x

in

U

, i.e. if

lim

z

→

z

0

⁡

f

⁢

(

z

)

-

f

⁢

(

z

0

)

z

-

z

0

exists for all

z

0

∈

U

.

More generally, if

Ω

⊂

ℂ

n

is a domain, then a function

f

:

Ω

→

ℂ

is said to be

holomorphic

if

f

is holomorphic in each of the variables. The class of all

holomorphic functions

on

Ω

is usually denoted by

𝒪

⁢

(

Ω

)

.

Title

holomorphic

Canonical name

Holomorphic

Date of creation

2013-03-22 12:04:33

Last modified on

2013-03-22 12:04:33

Owner

djao (24)

Last modified by

djao (24)

Numerical id

12

Author

djao (24)

Entry type

Definition

Classification

msc 30D20

Classification

msc 32A10

Synonym

holomorphic function

Synonym

regular function

Synonym

complex differentiable

Related topic

CauchyRiemannEquations

Related topic

Analytic

Generated on Fri Feb 9 20:18:00 2018 by

LaTeXML
Liouville’s theorem

Liouville’s theorem

A

bounded

entire function

is constant. That is, a bounded

complex function

f

:

ℂ

→

ℂ

which is

holomorphic

on the entire

complex plane

is always a constant function.

More generally, any

holomorphic function

f

:

ℂ

→

ℂ

which satisfies a

polynomial

bound condition of the form

|

f

⁢

(

z

)

|

<

c

⋅

|

z

|

n

for some

c

∈

ℝ

,

n

∈

ℤ

, and all

z

∈

ℂ

with

|

z

|

sufficiently large is necessarily equal to a polynomial function.

Liouville’s theorem is a vivid example of how stringent the holomorphicity condition on a complex function really is. One has only to compare the theorem to the corresponding statement for real functions (namely, that a bounded

differentiable

real function is constant, a patently false statement) to see how much stronger the complex differentiability condition is compared to real differentiability.

Applications of Liouville’s theorem include proofs of the

fundamental theorem of algebra

and of the

partial fraction decomposition

theorem for

rational functions

.

Title

Liouville’s theorem

Canonical name

LiouvillesTheorem

Date of creation

2013-03-22 12:04:31

Last modified on

2013-03-22 12:04:31

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Theorem

Classification

msc 30D20

Generated on Fri Feb 9 20:18:02 2018 by

LaTeXML
Morera’s theorem

Morera’s theorem

Morera’s

theorem

provides the

converse

of Cauchy’s

integral

theorem.

Theorem

[

1

]

Suppose

G

is a region in

ℂ

, and

f

:

G

→

ℂ

is a

continuous
function

. If for every closed

triangle

Δ

in

G

, we have

∫

∂

⁡

Δ

f

⁢

𝑑

z

=

0

,

then

f

is

analytic

on

G

. (Here,

∂

⁡

Δ

is the

piecewise

linear

boundary

(

http://planetmath.org/BoundaryInTopology

) of

Δ

.)

In particular, if for every rectifiable closed curve

Γ

in

G

, we have

∫

Γ

f

⁢

𝑑

z

=

0

,

then

f

is analytic on

G

. Proofs of this can be found most
undergraduate books on complex analysis

[

2

,

3

]

.

References

1

W. Rudin,

Real and complex analysis

, 3rd ed., McGraw-Hill Inc., 1987.

2

E. Kreyszig,

Advanced Engineering Mathematics

,
John Wiley & Sons, 1993, 7th ed.

3

R.A. Silverman,

Introductory Complex Analysis

,
Dover Publications, 1972.

Title

Morera’s theorem

Canonical name

MorerasTheorem

Date of creation

2013-03-22 12:58:09

Last modified on

2013-03-22 12:58:09

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

12

Author

matte (1858)

Entry type

Theorem

Classification

msc 30D20

Generated on Fri Feb 9 20:18:03 2018 by

LaTeXML
proof of addition formula of exp

proof of addition formula of exp

The

addition formula

e

z

1

+

z

2

=

e

z

1

⁢

e

z

2

of the

complex exponential function

may be proven by applying Cauchy multiplication rule to the

Taylor series

expansions (

http://planetmath.org/TaylorSeries

) of the right side factors (

http://planetmath.org/

Product

).  We present a proof which is based on the

derivative

of the

exponential function

.

Let

a

be a

complex

constant

.  Denote

e

z

=

w

⁢

(

z

)

.  Then

w

′

⁢

(

z

)

≡

w

⁢

(

z

)

.  Using the

product rule

and the

chain rule

we calculate:

d

d

⁢

z

1

⁢

[

w

⁢

(

z

1

)

⁢

w

⁢

(

a

-

z

1

)

]

=

w

′

⁢

(

z

1

)

⁢

w

⁢

(

a

-

z

1

)

+

w

⁢

(

z

1

)

⁢

w

′

⁢

(

a

-

z

1

)

⁢

(

-

1

)

=

e

z

1

⁢

e

a

-

z

1

-

e

z

1

⁢

e

a

-

z

1

≡

0

Thus we see that the product

w

⁢

(

z

1

)

⁢

w

⁢

(

a

-

z

1

)

=

e

z

1

⁢

e

a

-

z

1

must be a constant

A

.  If we choose specially

z

1

=

0

,  we obtain:

A

=

w

⁢

(

0

)

⁢

w

⁢

(

a

-

0

)

=

e

0

⁢

e

a

=

e

a

Therefore

e

z

1

⁢

e

a

-

z

1

≡

e

a

.

If we denote

a

-

z

1

:=

z

2

, the preceding equation reads

e

z

1

⁢

e

z

2

=

e

z

1

+

z

2

. Q.E.D.

Title

proof of addition formula of exp

Canonical name

ProofOfAdditionFormulaOfExp

Date of creation

2013-03-22 16:32:03

Last modified on

2013-03-22 16:32:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Proof

Classification

msc 30D20

Related topic

AdditionFormula

Related topic

AdditionFormulas

Defines

addition formula of exponential function

Generated on Fri Feb 9 20:18:08 2018 by

LaTeXML
proof of extended Liouville’s theorem

proof of extended Liouville’s theorem

This is a proof of the second, more general, form of Liouville’s

theorem

given in the parent (

http://planetmath.org/LiouvillesTheorem2

) article.

Let

f

:

ℂ

→

ℂ

be a

holomorphic function

such that

|

f

⁢

(

z

)

|

<

c

⋅

|

z

|

n

for some

c

∈

ℝ

and for

z

∈

ℂ

with

|

z

|

sufficiently large. Consider

g

⁢

(

z

)

=

{

f

⁢

(

z

)

-

f

⁢

(

0

)

z

z

≠

0

f

′

⁢

(

0

)

z

=

0

Since

f

is holomorphic,

g

is as well, and by the bound on

f

, we have

|

g

⁢

(

z

)

|

<

c

1

+

c

2

⋅

|

z

|

n

-

1

<

c

′

⋅

|

z

|

n

-

1

again for

|

z

|

sufficiently large.

By

induction

,

g

is a polynomial of degree at most

n

-

1

, and thus

f

is a polynomial of degree at most

n

.

Title

proof of extended Liouville’s theorem

Canonical name

ProofOfExtendedLiouvillesTheorem

Date of creation

2013-03-22 16:18:31

Last modified on

2013-03-22 16:18:31

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

8

Author

rm50 (10146)

Entry type

Proof

Classification

msc 30D20

Generated on Fri Feb 9 20:18:10 2018 by

LaTeXML
proof of Liouville’s theorem

proof of Liouville’s theorem

Let

f

:

ℂ

→

ℂ

be a

bounded

,

entire function

. Then by Taylor’s theorem,

f

⁢

(

z

)

=

∑

n

=

0

∞

c

n

⁢

x

n

⁢

where

⁢

c

n

=

1

2

⁢

π

⁢

i

⁢

∫

Γ

r

f

⁢

(

w

)

w

n

+

1

⁢

𝑑

w

where

Γ

r

is the circle of radius

r

about

0

, for

r

>

0

. Then

c

n

can be estimated as

|

c

n

|

≤

1

2

⁢

π

⁢

length

⁡

(

Γ

r

)

⁢

sup

⁡

{

|

f

⁢

(

w

)

w

n

+

1

|

:

w

∈

Γ

r

}

=

1

2

⁢

π

⁢

2

⁢

π

⁢

r

⁢

M

r

r

n

+

1

=

M

r

r

n

where

M

r

=

sup

⁡

{

|

f

⁢

(

w

)

|

:

w

∈

Γ

r

}

.

But

f

is bounded, so there is

M

such that

M

r

≤

M

for all

r

. Then

|

c

n

|

≤

M

r

n

for all

n

and all

r

>

0

. But since

r

is arbitrary, this gives

c

n

=

0

whenever

n

>

0

. So

f

⁢

(

z

)

=

c

0

for all

z

, so

f

is constant.

□

Title

proof of Liouville’s theorem

Canonical name

ProofOfLiouvillesTheorem

Date of creation

2013-03-22 12:54:15

Last modified on

2013-03-22 12:54:15

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

5

Author

Evandar (27)

Entry type

Proof

Classification

msc 30D20

Generated on Fri Feb 9 20:18:12 2018 by

LaTeXML
proof of Morera’s theorem

proof of Morera’s theorem

We provide a proof of Morera’s

theorem

under the

hypothesis

that

∫

Γ

f

⁢

(

z

)

⁢

𝑑

z

=

0

for any

circuit

Γ

contained

in

G

.
This is apparently more restrictive, but actually

equivalent

,
to supposing

∫

∂

⁡

Δ

f

⁢

(

z

)

⁢

𝑑

z

=

0

for any

triangle

Δ

⊆

G

,
provided that

f

is

continuous

in

G

.

The idea is to prove that

f

has an antiderivative

F

in

G

.
Then

F

, being holomorphic in

G

,
will have

derivatives

of any order in

G

;
but

F

(

n

)

⁢

(

z

)

=

f

(

n

-

1

)

⁢

(

z

)

for all

z

∈

G

,

n

∈

ℕ

,

n

≥

1

.

First, suppose

G

is

connected

.
Then

G

, being open, is also

pathwise connected

.

Fix

z

0

∈

G

.
For any

z

∈

G

define

F

⁢

(

z

)

as

F

⁢

(

z

)

=

∫

γ

⁢

(

z

0

,

z

)

f

⁢

(

w

)

⁢

𝑑

w

,

(1)

where

γ

⁢

(

z

0

,

z

)

is a path entirely contained in

G

with initial point

z

0

and final point

z

.

The

function

F

:

G

→

ℂ

is well defined.
In fact, let

γ

1

and

γ

2

be any two paths entirely contained in

G

with initial point

z

0

and final point

z

;
define a circuit

Γ

by joining

γ

1

and

-

γ

2

,
the path obtained from

γ

2

by
“reversing the

parameter

direction”.
Then by linearity and

additivity

of integral

∫

Γ

f

⁢

(

w

)

⁢

𝑑

w

=

∫

γ

1

f

⁢

(

w

)

⁢

𝑑

w

+

∫

-

γ

2

f

⁢

(

w

)

⁢

𝑑

w

=

∫

γ

1

f

⁢

(

w

)

⁢

𝑑

w

-

∫

γ

2

f

⁢

(

w

)

⁢

𝑑

w

;

(2)

but the left-hand side is 0 by hypothesis,
thus the two integrals on the right-hand side are equal.

We must now prove that

F

′

=

f

in

G

.
Given

z

∈

G

, there exists

r

>

0

such that the ball

B

r

⁢

(

z

)

of radius

r

centered in

z

is contained in

G

.
Suppose

0

<

|

Δ

⁢

z

|

<

r

:
then we can choose as a path from

z

to

z

+

Δ

⁢

z

the

segment

γ

:

[

0

,

1

]

→

G

parameterized by

t

↦

z

+

t

⁢

Δ

⁢

z

.
Write

f

=

u

+

i

⁢

v

with

u

,

v

:

G

→

ℝ

:
by additivity of integral and the mean value theorem,

F

⁢

(

z

+

Δ

⁢

z

)

-

F

⁢

(

z

)

Δ

⁢

z

=

1

Δ

⁢

z

⁢

∫

γ

f

⁢

(

w

)

⁢

𝑑

w

=

1

Δ

⁢

z

⁢

∫

0

1

f

⁢

(

z

+

t

⁢

Δ

⁢

z

)

⁢

Δ

⁢

z

⁢

𝑑

t

=

u

⁢

(

z

+

θ

u

⁢

Δ

⁢

z

)

+

i

⁢

v

⁢

(

z

+

θ

v

⁢

Δ

⁢

z

)

for some

θ

u

,

θ

v

∈

(

0

,

1

)

.
Since

f

is continuous, so are

u

and

v

, and

lim

Δ

⁢

z

→

0

⁡

F

⁢

(

z

+

Δ

⁢

z

)

-

F

⁢

(

z

)

Δ

⁢

z

=

u

⁢

(

z

)

+

i

⁢

v

⁢

(

z

)

=

f

⁢

(

z

)

.

In the general case, we just repeat the procedure
once for each

connected component

of

G

.

Title

proof of Morera’s theorem

Canonical

name

ProofOfMorerasTheorem

Date of creation

2013-03-22 18:53:34

Last modified on

2013-03-22 18:53:34

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

10

Author

Ziosilvio (18733)

Entry type

Proof

Classification

msc 30D20

Generated on Fri Feb 9 20:18:15 2018 by

LaTeXML
properties of entire functions

properties of entire functions

1.

If

f

:

ℂ

→

ℂ

is an

entire function

and

z

0

∈

ℂ

, then

f

⁢

(

z

)

has the

Taylor series

f

⁢

(

z

)

=

a

0

+

a

1

⁢

(

z

-

z

0

)

+

a

2

⁢

(

z

-

z

0

)

2

+

⋯

which is valid in the whole

complex plane

.

2.

If, conversely, such a

power series

converges for every complex value

z

, then the sum of the series (

http://planetmath.org/SumFunctionOfSeries

) is an entire function.

3.

The entire functions may be divided in two disjoint :

a) The

entire rational functions

, i.e. polynomial functions; in their series there is an

n

0

such that

a

n

=

0

⁢

∀

n

≧

n

0

.

b) The

entire transcendental functions

; in their series one has

a

n

≠

0

for infinitely many values of

n

.  Examples are

complex sine and cosine

,

complex exponential function

,

sine integral

,

error function

.

4.

A consequence of Liouville’s theorem:  If

f

is a non-constant entire function and if

R

and

M

are two arbitrarily great positive numbers, then there exist such points

z

that

|

z

|

>

R

⁢

and

⁢

|

f

⁢

(

z

)

|

>

M

.

This that the non-constant entire functions are unbounded (

http://planetmath.org/BoundedFunction

).

5.

The sum (

http://planetmath.org/SumOfFunctions

), the product (

http://planetmath.org/ProductOfFunctions

) and the

composition

of two entire functions are entire functions.

6.

The ring of all entire functions is a Prüfer domain.

References

1

O. Helmer:

“Divisibility properties of integral functions”.  –

Duke Math. J.

6

(1940), 345–356.

Title

properties of entire functions

Canonical name

PropertiesOfEntireFunctions

Date of creation

2013-03-22 14:52:09

Last modified on

2013-03-22 14:52:09

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Result

Classification

msc 30D20

Related topic

RationalFunction

Related topic

AlgebraicFunction

Related topic

BesselsEquation

Defines

entire rational function

Defines

entire transcendental function

Generated on Fri Feb 9 20:18:17 2018 by

LaTeXML