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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2010 • Question 6

Question

Math 6210 • Real Analysis Spring 2010 • Question 6

Let \[ f(z)=\frac {6z+2}{(z^2-1)(z+3)}. \]

(a) Use the residue theorem to compute \(\int _\gamma f(z)\,dz\), where \(\gamma \) is the positively oriented circle \(|z|=2\).

(b) Compute the Laurent series for \(f(z)\) which converges in the annulus \(1<|z|<3\). Use this to check your computation of the integral in (a).

Raw LaTeX

Let
\[
f(z)=\frac{6z+2}{(z^2-1)(z+3)}.
\]

(a) Use the residue theorem to compute \(\int_\gamma f(z)\,dz\), where
\(\gamma\) is the positively oriented circle \(|z|=2\).

(b) Compute the Laurent series for \(f(z)\) which converges in the annulus
\(1<|z|<3\). Use this to check your computation of the integral in (a).

Topic Tags

Cauchy Integral Theory and Morera's Theorem Isolated Singularities and Residues Laurent Series and Annuli Lebesgue Integration