Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Class 20 (Wednesday March 4)

SUMMARY Introduction to Contour Integration CURRENT READING Brown & Curchill, pages 95-104 NEXT READING Brown & Curchill pages 104-122

\underline{Exercise} First, let’s recall how to integrate complex functions of a real variable. Compute the following:

(a)∫_{1}^{2} \frac{−i}{t^{2}} + (t + 2i)^{3} dt (b)∫_{0}^{∞} e^{−z^{2}t} dt

Contour Integration Integration of a complex function of a complex variable is performed on a set of connected points from, say, z_{1} to z_{2}. It is a contour integral. Given a contour C defined as z(t) for a ≤ t ≤ b where z_{1} = z(a) and z_{2} = z(b), an integral of a complex function of a complex variable f(z) is written as

∫_{C} f(z)dz  or  ∫_{z_{1}}^{z_{2}} f(z) dz

Let f(z) be piecewise continuous on z(t). If C is a contour then z′(t) is piecewise continuous on a ≤ t ≤ b and we can redefine the integral of f(z) along C as:

∫_{C} f(z) dz = ∫_{a}^{b} f[z(t)]z′(t) dt

\underline{Examples}

Compute ∫_{C} Re z dz where C is a directed line segment from z = 0 to z = 1 + 2i ALGORITHM: steps to be taken to complete the process 1: write down a parametricization for C, z(t) 2: Convert the integral into an integral in (real) t variables 3: Integrate!

\underline{Groupwork} Compute ∫_{C} 2\overline{z}^{2} dz where C is a directed l ine segment from z = 2 to z = −2. (Sketch the contour and evaluate the integral.)

Also ∫_{C} 2\overline{z}^{2} dz, this time using C being a circular arc from z = 2 to z = −2. (Sketch the contour and then evaluate the integral.)

\underline{Exercise} Show that

∫_{C_{r}}(z − z_{0})^{n} dz = \{ \begin{matrix}2πi & n = −1 \\ 0 & n ≠ −1 \end{matrix}

where n is any integer and C_{r} is a circle of radius r around z_{0} (what is the equation of such a shape?) traversed once in the counter-clockwise direction