Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Does there exist an explicit formula for the sum of the series $$ \sum_{n=1}^\infty \frac{1}{n^2-z^2}? $$

share|cite|improve this question
Yes (the question is often asked here). See for example this. – Raymond Manzoni Dec 19 '12 at 20:42
That's very nice, many thanks! (The answer is accepted, so I hope it contains a correct proof.) – deltuva Dec 19 '12 at 20:45
Thanks, of course Eric's answer is correct? Another link with links... – Raymond Manzoni Dec 19 '12 at 20:46
up vote 1 down vote accepted

Equation $(18)$ on this page states that $$\pi \cot(\pi z)=\frac{1}{z}+2z\sum_{n=1}^\infty \frac{1}{z^2-n^2}.$$

share|cite|improve this answer

This answer handles precisely this question by showing that the Cauchy Principal Value of $$ \sum_{k\in\mathbb{Z}}\frac1{k+z}=\pi\cot(\pi z) $$ and derives an explicit value for $\displaystyle\sum_{k=1}^\infty\frac{1}{k^2-z^2}$ in $(9)$.

share|cite|improve this answer

Yes, there does. Of course the sum is not defined if $z$ is a nonzero integer.

share|cite|improve this answer

Your Answer


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.