Complex Analysis

Math 312  MWF 10:30am - 11:25am
\begin{matrix}c \\ ◯ \end{matrix}1998 Ron Buckmire  Fowler 112

Class 30 (Wednesday April 8)

SUMMARY Using Laurent Series to Compute Residues CURRENT READING Brown & Curchill pages 154�158 NEXT READING Brown & Curchill pages 210�219

Laurent Series

f(z) = ∑_{n=0}^{∞}a_{n}(z − z_{0})^{n} + ∑_{n=1}^{∞} \frac{b_{n}}{(z − z_{0})^{n}} , R_{1} < |z − z_{0}| < R_{2}

This formula for a Laurent series is also sometimes written as

f(z) = ∑_{n=−∞}^{∞} c_{n}(z − z_{0})^{n} where c_{n} = \frac{1}{2πi} ∮_{C} \frac{f(z)}{(z − z_{0})^{n+1}}dz, n = 0, ±1, ±2, …

In practice, one usually computes a Laurent series by comparing the function you have to one of the �famous functions� whose Maclaurin series you already know. \underline{Groupwork}

1. Write down the Laurent series for z^{2} sin(1/z^{2}) in the domain 0 < |z| < ∞

2. What is the value of Res(z^{2} sin(1/z^{2}), 0)?

3. Evaluate ∮_{|z|=1} z^{2} sin(1/z^{2})dz 4. Write down two different Laurent series for f(z) = \frac{1}{z(z^{2} + 1)} and specify the domain in which your series are valid.

5. Evaluate ∮_{|z|=2} \frac{1}{z(z^{2} + 1)}dz using whichever method you prefer.