# Evaluating Integrals Using Cauchy's Formula/Theorem

1. Mar 4, 2008

### e(ho0n3

[SOLVED] Evaluating Integrals Using Cauchy's Formula/Theorem

1. The problem statement, all variables and given/known data
Evaluate the given integral using Cauchy's Formula or Theorem.

$$\int_{|z+1|=2} \frac{z^2}{4 - z^2} \, dz$$

2. Relevant equations
Cauchy's Theorem. Suppose that f is analytic on a domain D. Let $\gamma$ be a piecewise smooth simple closed curve in D whose inside $\Omega$ also lies in D. Then

$$\int_\gamma f(z) \, dz = 0$$

Cauchy's Formula. Suppose that f is analytic on a domain D and $\gamma$ is a piecewise smooth, positively oriented simple closed curve D whose inside $\Omega$ also lies in D. Then

$$f(z) = \frac{1}{2\pi i} \int_\gamma \frac{f(\zeta)}{\zeta - z} \, d\zeta$$

for all $z \in \Omega$.

3. The attempt at a solution
For z equal to 2 and -2, the integrand is not defined and hence is not analytic in any domain that includes these points. The problem is that the curve |z + 1| = 2 contains the point -2 inside it so I can't apply either Cauchy's Theorem or Formula. What can I do?

2. Mar 4, 2008

### ircdan

hint

z^2/(4 - z^2) = z^2/((2 - z)(2 + z)) = (z^2/(z-2))/(z+2) = g(z)/(z - (-2)), where g(z) = z^2/(z-2) is analytic inside and on your circle

3. Mar 4, 2008

### e(ho0n3

Ah! Thanks for the tip. That really helps.