How to prove this formula

Question

I want to know how to prove this:

$$ \sum_{k=1}^\infty \arctan\left( \frac{x}{k^2+a} \right) = \pi \left\lfloor \frac{b}{\sqrt{2}} \right\rfloor + \arctan\left( \frac{x}{b^2} \right) - \arctan\left[ \tanh\left( \frac{\pi}{\sqrt{2}} \frac{x}{b} \right) \cot\left( \frac{\pi}{\sqrt{2}} b \right) \right] $$

for $0 \le a, b = \sqrt{\sqrt{x^2 + a^2} - a}$.

At least in the special case $x=1$,$a=0$

Thank you ..

See http://math.stackexchange.com/questions/613670/prove-this-sum-n-1-infty-arctan-left-dfrac1n21-right-arctan-l — user91500, Jan 11 '14 at 07:29
How is this a duplicate of that question? In particular kong asks for $$\sum_{k=1}^\infty\arctan\left(\frac1{k^2}\right)$$ which is asked in this question. — robjohn, Jan 11 '14 at 13:17

score 1 · Answer 1 · edited Apr 13 '17 at 12:21

Let $\;\displaystyle u + iv = \pi\sqrt{a+ix} = \frac{\pi}{\sqrt{2}}\left(\frac{x}{b} + ib\right)$ and follow nearly exactly the same step as in this answer, you get

$$\begin{align} \sum_{k=1}\tan^{-1}\left(\frac{x}{k^2+a}\right) &= \tan^{-1}\left(\frac{\tan v}{\tanh u}\right) - \tan^{-1}\left(\frac{v}{u}\right) + \pi N\\ &= \tan^{-1}\left(\frac{x}{b^2}\right) - \tan^{-1}\left(\tanh\left(\frac{\pi x}{\sqrt{2}{b}}\right)\cot\left(\frac{\pi b}{\sqrt{2}}\right)\right) + \pi N \end{align}$$ for some integer $N$. The only thing left is to figure out what $N$ is...

How to prove this formula

1 Answers1