topic entry on complex analysis

Introduction

Complex analysis may be defined as the study of analyticfunctions of a complex variable. The origins of this subjectlie in the observation that, given a function which has aconvergent Taylor series, one can substitute complex numbersfor the variable and obtain a convergent series which definesa function of a complex variable. Puttingimaginary numbers into the power series for the exponentialfunction, we find

	$\\displaystyle e^{ix}$	$\\displaystyle=$	$\\displaystyle 1+ix-\\frac{x^{2}}{2}-i\\frac{x^{3}}{3!}+\\frac{x^{4}}{4!}+i\\frac{x%^{5}}{5!}-\\frac{x^{6}}{6!}-i\\frac{x^{7}}{7!}+\\cdots$
	$\\displaystyle e^{-ix}$	$\\displaystyle=$	$\\displaystyle 1-ix-\\frac{x^{2}}{2}+i\\frac{x^{3}}{3!}+\\frac{x^{4}}{4!}-i\\frac{x%^{5}}{5!}-\\frac{x^{6}}{6!}+i\\frac{x^{7}}{7!}+\\cdots$

Adding and subtracting these series, we find

	$\\displaystyle\\frac{1}{2}(e^{ix}+e^{-ix})$	$\\displaystyle=$	$\\displaystyle 1-\\frac{x^{2}}{2!}+\\frac{x^{4}}{4!}-\\frac{x^{6}}{6!}+-\\cdots$
	$\\displaystyle\\frac{1}{2i}(e^{ix}-e^{-ix})$	$\\displaystyle=$	$\\displaystyle x-\\frac{x^{3}}{3!}+\\frac{x^{5}}{5!}-\\frac{x^{7}}{7!}+-\\cdots$

We recognize these series as the Taylor-Madhava series forthe sine and the cosine functions respectively. We hence have

$\\displaystyle\\sin x$	$\\displaystyle=$	$\\displaystyle\\frac{1}{2i}(e^{ix}-e^{-ix})$
$\\displaystyle\\cos x$	$\\displaystyle=$	$\\displaystyle\\frac{1}{2}(e^{ix}+e^{-ix})$
$\\displaystyle e^{ix}$	$\\displaystyle=$	$\\displaystyle\\cos x+i\\sin x.$

These equations let us re-express trigonometric functionsin terms of complex exponentials. Using them, deriving andverifying trigonometric identities becomes a straightforwardexercise in algebra using the laws of exponents.

We call functions of a complex variable which can beexpressed in terms of a power series as complexanalytic. More precisely, if $D$ is an opensubset of $\\mathbb{C}$ , we say that a function $f\\colon D\\to\\mathbb{C}$ is complex analyticif, for every point $w$ in $D$ , there exists a positivenumber $\\delta$ and a sequence of complex numbers $c_{k}$ such that the series

\\sum_{k=0}^{\\infty}c_{k}(z-w)^{k}

converges to $f(z)$ when $z\\in D$ and $|z-w|<\\delta$ .

An important feature of this definition is that itis not required that a single series works forall points of $D$ . For instance, supposewe define the function $f\\colon\\mathbb{C}\\!\\smallsetminus\\!\\{1\\}\\to\\mathbb{C}$ as

f(z)={1\\over 1-z}.

While it it turns out that $f$ is analytic, no singleseries will give us the values of $f$ for all allowedvalues of $z$ . For instance, we have the familiargeometric series:

f(z)=\\sum_{k=0}^{\\infty}z^{k}

However, this series diverges when $|z|>1$ . Forsuch values of $z$ , we need to use other series.For instance, when $z$ is near $2$ , we have thefollowing series:

f(z)=\\sum_{k=0}^{\\infty}(-1)^{k+1}(z-2)^{k}

This series, however, diverges when $|z-2|>1$ . While, for every allowed value of $z$ we can findsome power series which will converge to $f(z)$ , nosingle power series will converge to $f(z)$ forall permissible values of $z$ .

It is possible to define the operations ofdifferentiation and integration for complexfunctions. These operations are well-definedfor analytic functions and have the usualproperties familiar from real analysis.

The class of analytic functions is interestingto study for at least two main reasons. Firstly,many functions which arise in pure and appliedmathematics, such as polynomials, rational functions,exponential functions. logarithms, trigonometricfunctions, and solutions of differential equationsare analytic. Second, the class of analytic functionsenjoys many remarkable properties which do not holdfor other classes of functions, such as the following:

Closure: The class of complex analytic functionsis closed under the usual algebraic operations,taking derivative and integrals, composition,and taking uniform limits.
Rigidity: Given a complex analytic function $f\\colon D\\to\\mathbb{C}$ , where $D$ isan open subset of $\\mathbb{C}$ , if we know thevalues of $f$ at an infinite number of points of $D$ which have a limit point in $D$ , then weknow the value of $f$ at all points of $D$ . Forinstance, given a complex analytic function onsome neighborhood of the real axis, the valuesof that function in the whole neighborhood willbe determined by its values on the real axis.
Cauchy and Morera theorems: The integral of acomplex analytic function along any contractibleclosed loop equals zero. Conversely, if the integralof a complex function about every contractible loophappens to be zero, then that function is analytic.
Complex differentiability: If a complexfunction is differentiable, then it has derivativesof all orders. This contrasts sharply with thecase of real analysis, where a function may bedifferentiable only a fixed number of times.
Harmonicity: The real and imaginary partsof a complex analytic function are harmonic, i.e.satisfy Laplace’s equation. Conversely, given aharmonic function on the plane, there exists acomplex analytic function of which it is the realpart.
Conformal mapping: A complex function isanalytic if and only if it preserves maps pairsof intersecting curves into pairs which intersectat the same angle.

As one can see, there are many ways to characterizecomplex analytic functions, many of which havenothing to do with power series. This suggests thatanalytic functions are somehow a naturally occurringsubset of complex functions. This variety of distinctways of characterizing analytic functions means thatone has a variety of methods which may be used tostudy them and prove deep and surprising resultsby bringing insights and techniques from geometry,differential equations, and functional analysis tobear on problems of complex analysis. This alsoworks the other way — one can use complex analysisto prove results in other branches of mathmaticswhich have nothing to do with complex numbers. Forinstance, the problem of minimal surfaces can besolved 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 $\\mathbb{C}$
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/8190value of the Riemann zeta function at 0, http://planetmath.org/node/4719at 2, http://planetmath.org/node/11009at 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/6289Schwarz–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