Basics of Complex Analysis

This post contains material from the book titles “Complex Variables” by Stephen D. Fisher. The book is at an advanced level, contains excellent examples, and most importantly, emphasizes the different hypothesis for the theorems in complex analysis. I often feel that one gets lost in theorems that seem pretty similar, but are yet different. Instead of spending a lot of time trying to figure out yourself what the differences are, I think it is very very helpful to just know the differences between similar sounding theorems by the word of the author.

For the purposes of complex analysis, we define a domain to be an open and connected subset of $\mathbb{R}^2$ .

Green’s Theorem in the Plane

Green’s theorem is valid for a domain $\Omega$ whose boundary $\Gamma$ consists of a finite number of disjoint, piecewise smooth, simple, closed curves. Note that each adjective is important: 1) finite 2) disjoint 3) piecewise smooth 4) simple 5) closed. A curve $\gamma$ is a continuous function from some closed interval $[a,b]$ in the real line to the set of complex numbers. A curve is simple if the function is injective except that we allow $\gamma(a)=\gamma(b)$ . When $\gamma(a) = \gamma(b)$ , we call the curve closed. A curve $\gamma$ is smooth if $\gamma\,'$ exists and is continuous. The boundary $\Gamma$ is oriented so that the domain $\Omega$ remains to the left as we walk around $\Gamma$ .

Given functions $p$ and $q$ which are continuous and have continuous first partial-derivatives, then $\int_\Gamma p\,dx + q\,dy = \int_\Omega (q_x - p_y)\,dx\,dy$

Analyticity

Let $D$ be a a domain and $f$ be a function from $D$ to $\mathbb{C}$ . $f$ is said to be differentiable at a point $z_0\in D$ if the limit $(f(z)-f(z_0))/(z-z_0)$ exists as $z\rightarrow z_0$ . If $f$ is differentiable everywhere in $D$ , then $f$ is said to be analytic on $D$ . If $f$ is analytic on a domain $D$ , the Cauchy-Riemann equations $f_{z^*}=0$ hold throughout $D$ .

There is a converse to the Cauchy-Riemann equations. Suppose that $f= u+iv$ , that $u$ , $v$ and their first partial-derivatives are continuous in some open set about a point $z_0$ and that $u$ and $v$ satisfy the Cauchy-Riemann equations, then $f$ is differentiable at the point $z_0$ .

Power Series

A power series is an infinite series of the form $a_n(z-z_0)^n$ where $z_0$ and $a_n$ ‘s are complex numbers. Every such series has a radius of convergence $R$ , which can be zero or infinity, such that the series converges in the interior of the disk of radius $R$ centered at $z_0$ , and diverges in its exterior. No claim is made about convergence or divergence at any point of the boundary of this disk.

If a power series has a positive or infinite radius of curvature $R$ , we define a function $f$ from the domain $\vert z-z_0\vert < R$ by setting $f(z)$ to the number to which the series converges. We can show that $f$ has derivatives of all orders and each derivative can be obtained by termwise differentiation of the series for $f$ . It can be shown that the radii of convergence of the series obtained by termwise differentiation are atleast $R$ .

If we are given two power series centered at the same point, each with a radius of convergence atleast $R$ , their product series is well-defined inside the disk of convergence and is given by the usual product of series.

Cauchy-Goursat Theorem

If we assume that $f$ is analytic on a domain $D$ (that is, differentiable at each point of $D$ ), and that the first partial derivatives are continuous, then we can use Green’s Theorem to prove that the integral of $f$ is zero on any piecewise smooth, simple, closed curve in $D$ , whose inside also lies in $D$ . If we add the extra assumption that the domain $D$ is simply-connected, then we can show that the integral of $f$ is zero on any piecewise smooth, closed curve in $D$ , which may not necessarily be simple. Also, the assumption about $D$ being simply connected can be used to show that there is an analytic function $F$ on $D$ such that $F' = f$ throughout $D$ .

In 1900, Goursat discovered that the hypothesis $f\,'$ is continuous is superfluous in the above theorems. We can show this by first establishing that if $f$ is analytic in a domain $D$ , then $f$ has a valid power series in each disk that is contained in the domain.

Cauchy’s theorem also has a converse. If $f$ is continuous on a domain $D$ such that the integral of $f$ vanishes on every triangle that lies, together with its interior, in $D$ , then $f$ is analytic on $D$ . This is called Monera’s theorem.

Liouville’s theorem states that if a function $F$ is analytic on the whole complex plane, then its absolute value cannot be bounded, unless $F$ is a constant function.