PlanetMath: topic entry on complex analysis

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topic entry on complex analysis

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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. For instance, putting imaginary numbers into the power series for the exponential function, we find

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

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 for the 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 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 is an open subset of $\mathbb{C}$ , we say that a function $f \colon D \to \mathbb{C}$ is complex analytic if, for every point in , there exists a positive number $\delta$ and a sequence of complex numbers such that the series

$\displaystyle \sum_{k=0}^\infty c_k (z-w)^k$

converges to

when $z \in D$ and $\vert z - w\vert < \delta$ .

An important feature of this definition is that it is not required that a single series works for all points of . For instance, suppose we define the function $f \colon \mathbb{C}\!\smallsetminus\!\{1\} \to \mathbb{C}$ as

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

While it it turns out that

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

for all allowed values of

. For instance, we have the familiar geometric series:

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

However, this series diverges when $\vert z\vert > 1$ . For such values of

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

is near

, we have the following series:

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

This series, however, diverges when $\vert z-2\vert > 1$ . While, for every allowed value of

we can find some power series which will converge to

, no single power series will converge to

for all permissible values of

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 $f \colon D \to \mathbb{C}$ , where is an open subset of $\mathbb{C}$ , if we know the values of at an infinite number of points of which have a limit point in , then we know the value of at all points of . 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.