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1. The problem statement, all variables and given/known data

Evaluate the given integral using Cauchy's Formula or Theorem.

[tex]\int_{|z+1|=2} \frac{z^2}{4 - z^2} \, dz[/tex]

2. Relevant equations

Cauchy's Theorem. Suppose that f is analytic on a domain D. Let [itex]\gamma[/itex] be a piecewise smooth simple closed curve in D whose inside [itex]\Omega[/itex] also lies in D. Then

[tex]\int_\gamma f(z) \, dz = 0[/tex]

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

[tex]f(z) = \frac{1}{2\pi i} \int_\gamma \frac{f(\zeta)}{\zeta - z} \, d\zeta[/tex]

for all [itex]z \in \Omega[/itex].

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?

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# Evaluating Integrals Using Cauchy's Formula/Theorem

**Physics Forums | Science Articles, Homework Help, Discussion**