topic entry on complex analysis TopicEntryOnComplexAnalysis 2007-03-03 11:30:42 2009-10-15 19:49:35 Topic complex function complex analytic % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} \subsection*{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 \begin{eqnarray*} e^{ix} &=& 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 \\ e^{-ix} &=& 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 \end{eqnarray*} Adding and subtracting these series, we find \begin{eqnarray*} \frac{1}{2} (e^{ix} + e^{-ix}) &=& 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} +- \cdots \\ \frac{1}{2i} (e^{ix} - e^{-ix}) &=& x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} +- \cdots \end{eqnarray*} We recognize these series as the Taylor-Madhava series for the sine and the cosine functions respectively. We hence have \begin{eqnarray*} \sin x &=& \frac{1}{2i} (e^{ix} - e^{-ix}) \\ \cos x &=& \frac{1}{2} (e^{ix} + e^{-ix}) \\ e^{ix} &=& \cos x + i \sin x . \end{eqnarray*} 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 \emph{complex analytic}. More precisely, if $D$ 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 $w$ in $D$, there exists a positive number $\delta$ 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 it is not required that a single series works for all points of $D$. For instance, suppose we 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 single series will give us the values of $f$ for all allowed values of $z$. For instance, we have the familiar geometric series: \[ f(z) = \sum_{k=0}^\infty z^k \] 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: \[ f(z) = \sum_{k=0}^\infty (-1)^{k+1} (z-2)^k \] This series, however, diverges when\, $|z-2| > 1$.\, While, for every allowed value of $z$ we can 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: \begin{description} \item[Closure] The class of complex analytic functions is closed under the usual algebraic operations, taking derivative and integrals, composition, and taking uniform limits. \item[Rigidity] Given a complex analytic function\, $f \colon D \to \mathbb{C}$, where $D$ is an open subset of $\mathbb{C}$,\, 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. \item[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. \item[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. \item[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. \item[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. \end{description} 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. \subsection{Complex numbers} \begin{enumerate} \item complex plane, equality of complex numbers \item topology of the complex plane \item triangle inequality of complex numbers \item argument of product and quotient \item unit disc, annulus, closed complex plane \item \PMlinkname{$n$th root}{CalculatingTheNthRootsOfAComplexNumber} \item taking square root algebraically \item quadratic equation in $\mathbb{C}$ \item \PMlinkname{complex function}{ComplexFunction} \end{enumerate} \subsection{Complex functions} \begin{enumerate} \item de Moivre identity \item addition formula \item complex exponential function \item periodicity of exponential function \item complex sine and cosine \item values of complex cosine \item complex tangent and cotangent \item example of summation by parts \item Euler's formulas (see also \PMlinkname{this}{ComplexSineAndCosine}) \item complex logarithm \item general power \item fundamental theorems in complex analysis \item index of special functions \end{enumerate} \subsection{Analytic function} \begin{enumerate} \item holomorphic \item meromorphic \item periodic functions \item isolated singularity \item complex derivative \item Cauchy-Riemann equations \item \PMlinkname{power series}{PowerSeries} \item Bohr's theorem \item identity theorem of holomorphic functions \item Weierstrass double series theorem \item entire functions \item properties of entire functions \item \PMlinkname{pole of function}{Z_0IsAPoleOfF} \item zeros and poles of rational function \item when all singularities are poles \item Casorati-Weierstrass theorem \item Picard's theorem \item Laurent series \item coefficients of Laurent series \item residue \item regular at infinity \item Nevanlinna theory \end{enumerate} \subsection{Complex integration} \begin{enumerate} \item contour integral \item estimating theorem of contour integral \item theorems on complex function series \item Cauchy integral theorem \item Cauchy integral formula; variant of Cauchy integral formula \item \PMlinkname{residue theorem}{CauchyResidueTheorem} \item example of using residue theorem \item argument principle \item complex antiderivative \end{enumerate} \subsection{Analytic continuation} \begin{enumerate} \item analytic continuation \item meromorphic continuation \item analytic continuation by power series \item monodromy theorem \item Schwarz' reflection principle \item example of analytic continuation \item analytic continuation of gamma function \item analytic continuation of Riemann zeta to critical strip \item \PMlinkname{analytic continuation of Riemann zeta (using integral)}{AnalyticContinuationOfRiemannZetaUsingIntegral} \end{enumerate} \subsection{Riemann zeta function} \begin{enumerate} \item Riemann zeta function \item Euler product formula \item \PMlinkname{Riemann functional equation}{FunctionalEquationOfTheRiemannZetaFunction} \item critical strip \item \PMlinkid{value of the Riemann zeta function at 0}{8190},\, \PMlinkid{at 2}{4719},\, \PMlinkid{at 4}{11009} \item formulae for zeta in the critical strip \end{enumerate} \subsection{Conformal mapping} \begin{enumerate} \item conformal mapping \item conformal mapping theorem \item simple example of composed conformal mapping \item example of conformal mapping \item \PMlinkid{Schwarz--Christoffel transformation}{6289} \end{enumerate}