topic entry on complex analysis

topic entry on complex analysis

Introduction

Complex analysis

may be defined as the study of

analytic
functions

of a complex variable. The origins of this subject
lie in the observation that, given a

function

which has a

convergent

Taylor series

, one can substitute

complex numbers

for the variable and obtain a

convergent series

which defines
a function of a complex variable. Putting

imaginary numbers

into the

power series

for the

exponential
function

, we find

e

i

⁢

x

=

1

+

i

⁢

x

-

x

2

2

-

i

⁢

x

3

3

!

+

x

4

4

!

+

i

⁢

x

5

5

!

-

x

6

6

!

-

i

⁢

x

7

7

!

+

⋯

e

-

i

⁢

x

=

1

-

i

⁢

x

-

x

2

2

+

i

⁢

x

3

3

!

+

x

4

4

!

-

i

⁢

x

5

5

!

-

x

6

6

!

+

i

⁢

x

7

7

!

+

⋯

Adding and subtracting these series, we find

1

2

⁢

(

e

i

⁢

x

+

e

-

i

⁢

x

)

=

1

-

x

2

2

!

+

x

4

4

!

-

x

6

6

!

+

-

⋯

1

2

⁢

i

⁢

(

e

i

⁢

x

-

e

-

i

⁢

x

)

=

x

-

x

3

3

!

+

x

5

5

!

-

x

7

7

!

+

-

⋯

We recognize these series as the Taylor-Madhava series for
the sine and the

cosine functions

respectively. We hence have

sin

⁡

x

=

1

2

⁢

i

⁢

(

e

i

⁢

x

-

e

-

i

⁢

x

)

cos

⁡

x

=

1

2

⁢

(

e

i

⁢

x

+

e

-

i

⁢

x

)

e

i

⁢

x

=

cos

⁡

x

+

i

⁢

sin

⁡

x

.

These equations let us re-express

trigonometric functions

in terms of

complex

exponentials

. Using them, deriving and
verifying

trigonometric identities

becomes a straightforward
exercise in

algebra

using the laws of exponents.

We call functions of a complex variable which can be
expressed in terms of a power series as

complex
analytic

. More precisely, if

D

is an

open
subset

of

ℂ

, we say that a function

f

:

D

→

ℂ

is complex analytic
if, for every point

w

in

D

, there exists a positive
number

δ

and a

sequence

of complex numbers

c

k

such that the series

∑

k

=

0

∞

c

k

⁢

(

z

-

w

)

k

converges

to

f

⁢

(

z

)

when

z

∈

D

and

|

z

-

w

|

<

δ

.

An important feature of this definition is that it
is not required that a single series works for
all points of

D

. For instance, suppose
we define the function

f

:

ℂ

∖

{

1

}

→

ℂ

as

f

⁢

(

z

)

=

1

1

-

z

.

While it it turns out that

f

is analytic, no single
series will give us the values of

f

for all allowed
values of

z

. For instance, we have the familiar

geometric series

:

f

⁢

(

z

)

=

∑

k

=

0

∞

z

k

However, this series diverges when

|

z

|

>

1

.  For
such values of

z

, we need to use other series.
For instance, when

z

is near

2

, we have the
following series:

f

⁢

(

z

)

=

∑

k

=

0

∞

(

-

1

)

k

+

1

⁢

(

z

-

2

)

k

This series, however, diverges when

|

z

-

2

|

>

1

.
While, for every allowed value of

z

we can find
some power series which will converge to

f

⁢

(

z

)

, no
single power series will converge to

f

⁢

(

z

)

for
all permissible values of

z

.

It is possible to define the operations of

differentiation

and integration for

complex
functions

. These operations are well-defined
for analytic functions and have the usual
properties familiar from

real analysis

.

The class of analytic functions is interesting
to study for at least two main reasons. Firstly,
many functions which arise in pure and applied
mathematics, such as

polynomials

,

rational functions

,
exponential functions.

logarithms

, trigonometric
functions, and solutions of differential equations
are analytic. Second, the class of analytic functions
enjoys many remarkable properties which do not hold
for other classes of functions, such as the following:

Closure

The class of complex analytic functions
is closed under the usual algebraic operations,
taking

derivative

and

integrals

,

composition

,
and taking uniform limits.

Rigidity

Given a complex analytic function

f

:

D

→

ℂ

, where

D

is
an open subset of

ℂ

,  if we know the
values of

f

at an

infinite

number of points of

D

which have a

limit point

in

D

, then we
know the value of

f

at all points of

D

. For
instance, given a complex analytic function on
some

neighborhood

of the

real axis

, the values
of that function in the whole neighborhood will
be determined by its values on the real axis.

Cauchy and

Morera theorems

The integral of a
complex analytic function along any

contractible

closed loop equals zero. Conversely, if the integral
of a complex function about every contractible loop
happens to be zero, then that function is analytic.

Complex differentiability

If a complex
function is

differentiable

, then it has derivatives
of all orders. This contrasts sharply with the
case of real analysis, where a function may be
differentiable only a fixed number of times.

Harmonicity

The real and

imaginary parts

of a complex analytic function are

harmonic

, i.e.
satisfy Laplace’s equation. Conversely, given a

harmonic function

on the plane, there exists a
complex analytic function of which it is the

real
part

.

Conformal mapping

A complex function is
analytic if and only if it preserves maps pairs
of intersecting curves into pairs which intersect
at the same angle.

As one can see, there are many ways to characterize
complex analytic functions, many of which have
nothing to do with power series. This suggests that
analytic functions are somehow a naturally occurring
subset of complex functions. This variety of distinct
ways of characterizing analytic functions means that
one has a variety of methods which may be used to
study them and prove deep and surprising results
by bringing insights and techniques from geometry,
differential equations, and

functional analysis

to
bear on problems of complex analysis. This also
works the other way — one can use complex analysis
to prove results in other branches of mathmatics
which have nothing to do with complex numbers. For
instance, the problem of

minimal surfaces

can be
solved by using complex analysis.

0.1

Complex numbers

1.

complex plane

,

equality of complex numbers

2.

topology

of the complex plane

3.

triangle inequality of complex numbers

4.

argument of product and quotient

5.

unit disc

,

annulus

,

closed complex plane

6.

n

th root (

http://planetmath.org/CalculatingTheNthRootsOfAComplexNumber

)

7.

taking

square root

algebraically

8.

quadratic equation

in

ℂ

9.

complex function (

http://planetmath.org/ComplexFunction

)

0.2

Complex functions

1.

de Moivre identity

2.

addition formula

3.

complex exponential function

4.

periodicity

of exponential function

5.

complex sine and cosine

6.

values of complex cosine

7.

complex tangent and cotangent

8.

example of

summation by parts

9.

Euler’s formulas (see also this (

http://planetmath.org/ComplexSineAndCosine

))

10.

complex logarithm

11.

general power

12.

fundamental theorems in complex analysis

13.

index of special functions

0.3

Analytic function

1.

holomorphic

2.

meromorphic

3.

periodic functions

4.

isolated singularity

5.

complex derivative

6.

Cauchy-Riemann equations

7.

power series (

http://planetmath.org/PowerSeries

)

8.

Bohr’s theorem

9.

identity theorem of holomorphic functions

10.

Weierstrass double series theorem

11.

entire functions

12.

properties of entire functions

13.

pole of function (

http://planetmath.org/Z_0IsAPoleOfF

)

14.

zeros and poles of rational function

15.

when all

singularities

are poles

16.

Casorati-Weierstrass theorem

17.

Picard’s theorem

18.

Laurent series

19.

coefficients of Laurent series

20.

residue

21.

regular at infinity

22.

Nevanlinna theory

0.4

Complex integration

1.

contour integral

2.

estimating theorem of contour integral

3.

theorems on complex function series

4.

holomorphic function associated with continuous function

5.

Cauchy integral theorem

6.

Cauchy integral formula

;

variant of Cauchy integral formula

7.

residue theorem

(

http://planetmath.org/CauchyResidueTheorem

)

8.

example of using residue theorem

9.

argument principle

10.

complex

antiderivative

0.5

Analytic continuation

1.

analytic continuation

2.

meromorphic continuation

3.

analytic continuation by power series

4.

monodromy theorem

5.

Schwarz’

reflection principle

6.

example of analytic continuation

7.

analytic continuation of gamma function

8.

analytic continuation of Riemann zeta to

critical strip

9.

analytic continuation of Riemann zeta (using integral) (

http://planetmath.org/AnalyticContinuationOfRiemannZetaUsingIntegral

)

0.6

Riemann zeta function

1.

Riemann zeta function

2.

Euler product formula

3.

Riemann functional equation (

http://planetmath.org/FunctionalEquationOfTheRiemannZetaFunction

)

4.

critical strip

5.

http://planetmath.org/node/8190

value of the Riemann zeta function at 0,

http://planetmath.org/node/4719

at 2,

http://planetmath.org/node/11009

at 4

6.

formulae for zeta in the critical strip

0.7

Conformal mapping

1.

conformal mapping

2.

conformal mapping theorem

3.

simple example of composed conformal mapping

4.

example of conformal mapping

5.

http://planetmath.org/node/6289

Schwarz–Christoffel

transformation

Title

topic entry on complex analysis

Canonical name

TopicEntryOnComplexAnalysis

Date of creation

2013-05-20 18:11:35

Last modified on

2013-05-20 18:11:35

Owner

pahio (2872)

Last modified by

unlord (1)

Numerical id

60

Author

pahio (1)

Entry type

Topic

Classification

msc 30A99

Related topic

HarmonicConjugateFunction

Related topic

TakingSquareRootAlgebraically

Related topic

CalculatingTheNthRootsOfAComplexNumber

Related topic

FundamentalTheoremOfAlgebra

Related topic

FundamentalTheoremsInComplexAnalysis

Defines

complex analytic

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