Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $f(z)$ be a rational function on $\mathbb{C}$. If the residues of $f$ at $z=0$ and $z=\infty$ are both $0$, is it true that $\oint_{\gamma} f(z)\mathrm dz=0$ ($\gamma$ is a closed curve in $\mathbb{C}$)? Thanks.

share|improve this question

2 Answers 2

up vote 6 down vote accepted

Unless I'm misunderstanding your question, $f(z) = \frac{1}{z-1}-\frac{1}{z+1}$ is a counterexample.

share|improve this answer

In fact, to have $\oint_\gamma f(z)\ dz =0$ for all closed contours $\gamma$ that don't pass through a pole of $f$, what you need is that the residues at all the poles of $f$ are 0. Since the sum of the residues at all the poles (including $\infty$) is always 0, it's necessary and sufficient that the residues at all the finite poles are 0.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.