nice-integral-dx-x-2-pi-2-cosh-x-

Question Number 142545 by mnjuly1970 last updated on 02/Jun/21

n i c e \dots . . i n t e g r a l

Ω := \int_{- \infty}^{+ \infty} \frac{d x}{(x^{2} + π^{2}) c o s h (x)} = ?

\dots . .

Answered by mindispower last updated on 02/Jun/21

= \int_{- \infty}^{\infty} \frac{2 d x}{(x^{2} + π^{2}) (e^{x} + e^{- x})} = \int_{- \infty}^{\infty} f (x) d x

e^{x} + e^{- x} = 0 \Rightarrow e^{2 x} = - 1 \Rightarrow x_{k} = (i k π + i \frac{π}{2}), k \in Z

Ω = 2 i π (\underset{k = 0}{\sum^{\infty}} R e s (f, x_{k}) + R e s (f, i π))

= 2 i π (\underset{k = 0}{\sum^{\infty}} \frac{1}{(- π^{2} {(k + \frac{1}{2})}^{2} + π^{2}) ({(- 1)}^{k} i} + \frac{2}{2 i π (- 2)}

- 1 - \frac{2}{π} \underset{0}{\sum^{\infty}} \frac{{(- 1)}^{k}}{{(k + \frac{1}{2})}^{2} - 1}

= - 1 - \frac{2}{π} (\underset{k = 0}{\sum^{\infty}} \frac{{(- 1)}^{k}}{(k - \frac{1}{2}) (k + \frac{3}{2})}

- 1 - \frac{1}{π} \sum_{k ⩾ 0} \frac{{(- 1)}^{k}}{(k - \frac{1}{2})} + \frac{1}{π} \sum_{k ⩾ 0} \frac{{(- 1)}^{k}}{k + \frac{3}{2}}

- 1 - \frac{1}{π} (- 2 - 2 + \sum_{k ⩾ 2} \frac{{(- 1)}^{k}}{k - \frac{1}{2}}) + \frac{1}{π} \sum_{k ⩾ 0} \frac{{(- 1)}^{k}}{k + \frac{3}{2}}

f i r s t s u m k \to k + 2

w e g e t Ω = - 1 + \frac{4}{π} - \frac{1}{π} {\sum_{k ⩾ 0} \frac{{(- 1)}^{k + 2}}{(k + 2 - \frac{1}{2})} - \sum_{k ⩾ 0} \frac{{(- 1)}^{k}}{k + \frac{3}{2}}}

= - 1 + \frac{4}{π}

\int_{- \infty}^{\infty} \frac{d x}{c h (x) (x^{2} + π^{2})} = - 1 + \frac{4}{π}