How to prove $\oint \frac{\pi\csc(πz)}{(2z+1) \cosh\left(\frac{\pi}{2} (2z+1)\right) } \,\mathrm dz ≈ 0$

Asked Dec 12 '14 at 06:11

Active Dec 12 '14 at 06:27

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In proving the $\displaystyle\frac{200}{\pi} \sum_{k=0}^\infty\frac{(-1)^k}{(2k+1) \cosh\left((2k+1) \frac\pi2\right) }= 25$ I use the below integral, because in calculation the integral by the residue theorem, the series appear and I can evaluate the amount of the series:

$$\oint\frac{\pi \csc(\pi z)}{(2z+1) \cosh\left(\frac\pi2 (2z+1)\right) } \,\mathrm dz $$

I mean by solving the below equation I can find the amount of the series:

$$\oint\frac{\pi \csc(\pi z)}{(2z+1) \cosh\left(\frac\pi2 (2z+1)\right) } \,\mathrm dz =2\pi i \left(- \frac \pi2+ 4\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1) \cosh\left((2n+1) \frac\pi2\right)}\right) $$

I cannot understand why the left tends to $0$, As $n\to\infty$

edited Dec 12 '14 at 06:27

Aditya Hase

8,851

asked Dec 12 '14 at 06:11

Fereshteh

1

Welcome to MSE. Please take a look at the Mathjax tutorial. – Null Dec 12 '14 at 06:16
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Proving $\frac{200}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)\cosh\left(\frac{\pi}{2}(2n+1)\right)}=25$ – Aditya Hase Dec 12 '14 at 06:37
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@ integrator thanks for your editting. In The Link, Just mention that it tends to 0, but there is no proof! – Fereshteh Dec 12 '14 at 06:44
Hint : Use $ML$ property $$\left|\int_C f(z); dz\right|\le ML$$ – Venus Dec 12 '14 at 09:23
@ Venus please explain more or link me to a website. I didnot get what youmean. – Fereshteh Dec 12 '14 at 09:51

How to prove $\oint \frac{\pi\csc(πz)}{(2z+1) \cosh\left(\frac{\pi}{2} (2z+1)\right) } \,\mathrm dz ≈ 0$

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