existence of power series

existence of power series

In this entry we shall demonstrate the

logical equivalence

of the

holomorphic

and

analytic

concepts

. As is the case with so many basic results in

complex analysis

, the proof of these facts hinges on the

Cauchy
integral theorem

, and the

Cauchy integral formula

.

Holomorphic implies analytic.

Theorem 1

Let

U

⊂

C

be an open domain that contains the origin, and
let

f

:

U

→

C

,

be a

function

such that
the

complex derivative

f

′

⁢

(

z

)

=

lim

ζ

→

0

⁡

f

⁢

(

z

+

ζ

)

-

f

⁢

(

z

)

ζ

exists for all

z

∈

U

. Then, there exists a

power series

representation

f

⁢

(

z

)

=

∑

k

=

0

∞

a

k

⁢

z

k

,

∥

z

∥

<

R

,

a

k

∈

ℂ

for a
sufficiently small

radius of convergence

R

>

0

.

Note: it is just as
easy to show the

existence

of a power series representation around
every

basepoint

in

z

0

∈

U

; one need only consider the

holomorphic
function

f

⁢

(

z

-

z

0

)

.

Proof.

Choose an

R

>

0

sufficiently small so that the
disk

∥

z

∥

≤

R

is

contained

in

U

.
By the Cauchy integral formula we have
that

f

⁢

(

z

)

=

1

2

⁢

π

⁢

i

⁢

∮

∥

ζ

∥

=

R

f

⁢

(

ζ

)

ζ

-

z

⁢

𝑑

ζ

,

∥

z

∥

<

R

,

where, as usual, the integration

contour

is oriented
counterclockwise. For every

ζ

of

modulus

R

, we can expand
the integrand as a geometric power series in

z

, namely

f

⁢

(

ζ

)

ζ

-

z

=

f

⁢

(

ζ

)

/

ζ

1

-

z

/

ζ

=

∑

k

=

0

∞

f

⁢

(

ζ

)

ζ

k

+

1

⁢

z

k

,

∥

z

∥

<

R

.

The circle of radius

R

is a

compact set

; hence

f

⁢

(

ζ

)

is

bounded

on it; and hence, the power series above

converges uniformly

with respect to

ζ

. Consequently, the order
of the

infinite

summation and the integration

operations

can be
interchanged. Hence,

f

⁢

(

z

)

=

∑

k

=

0

∞

a

k

⁢

z

k

,

∥

z

∥

<

R

,

where

a

k

=

1

2

⁢

π

⁢

i

⁢

∮

∥

ζ

∥

=

R

f

⁢

(

ζ

)

ζ

k

+

1

,

as desired. QED

Analytic implies holomorphic.

Theorem 2

Let

f

⁢

(

z

)

=

∑

n

=

0

∞

a

n

⁢

z

n

,

a

n

∈

ℂ

,

∥

z

∥

<

ϵ

be a power series, converging in

D

=

D

ϵ

⁢

(

0

)

, the

open disk

of radius

ϵ

>

0

about the origin. Then the complex derivative

f

′

⁢

(

z

)

=

lim

ζ

→

0

⁡

f

⁢

(

z

+

ζ

)

-

f

⁢

(

z

)

ζ

exists for all

z

∈

D

, i.e. the function

f

:

D

→

C

is holomorphic.

Note: this

theorem

generalizes immediately to shifted power series in

z

-

z

0

,

z

0

∈

ℂ

.

Proof.

For every

z

0

∈

D

, the function

f

⁢

(

z

)

can be recast
as a power series centered at

z

0

. Hence, without loss of
generality it suffices to prove the theorem for

z

=

0

. The power
series

∑

n

=

0

∞

a

n

+

1

⁢

ζ

n

,

ζ

∈

D

converges

, and equals

(

f

⁢

(

ζ

)

-

f

⁢

(

0

)

)

/

ζ

for

ζ

≠

0

. Consequently, the complex
derivative

f

′

⁢

(

0

)

exists; indeed it is equal to

a

1

. QED

Title

existence of power series

Canonical name

ExistenceOfPowerSeries

Date of creation

2013-03-22 12:56:27

Last modified on

2013-03-22 12:56:27

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Result

Classification

msc 30B10

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