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.

The theorem in its entirety is as follows:

Let $a_1,\ldots,a_l\in\mathbb{C}$ be pairwise different non-integral numbers. Let f be an analytic function in $\mathbb{C}-\{a_1,\ldots,a_l\}$ and set $g(z):=\pi \cot(\pi z)f(z)$, such that $|z^2f(z)|$ is bounded outside a suitable compact set. Then:

$$\sum_{n=-\infty}^\infty f(n) = -\sum_{j=1}^l \operatorname{Res}(g;a_j)$$

The book wants me to use this theorem to prove that $\sum_{n=1}^\infty \frac{1}{n^{2}} = \frac{\pi^2}{6}$. Everything points to me setting $f(z)=\frac{1}{z^2}$, but the pole of $\frac{1}{z^2}$ is zero which is an integral number and is thus a point at which f must be analytic, thus the theorem cannot be applied, what am I missing here? Thanks.

Edit: I guess I'm suppose to slightly modify the theorem so it works for an overlapping pole, I probably don't need clarification on this after all.

share|improve this question
add comment

1 Answer

Hint: you are on the right track. Try the function $$f(z) = \frac{1}{z^2 +a^2}$$ which coincides with your guess in the limit $a\to 0$. The additional term in the sum (due to the pole of $g(z)$ at $z=0$ can be simply subtracted). The rest of $\sum_n f(n)$ is two times the requested sum. Take the limit $a\to0$ and you are done...

share|improve this answer
add comment

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.