3

Evaluate

$$I(a)=\int\limits_0^{\infty}\frac{1}{{\cosh}({\pi}x)(1+ax^2)}dx\;,\;\;\text{with}\;\;a>0$$

For example, if $a=4$,then$I(4)=\frac{1}{2}\log(2)$.

Roger209
  • 1,233
  • 1
    do you want to evaluate the integral ??? – what'sup Oct 18 '13 at 13:35
  • 2
    "Show that"? Did you mean "evaluate"? What've you done so far, though? – DonAntonio Oct 18 '13 at 13:44
  • Ooo, these are my favorite kind of integrals. You mind sharing where the question came from? – David H Oct 18 '13 at 13:53
  • 1
    look at this too http://math.stackexchange.com/questions/411058/evaluate-the-integral-int-0-infty-frac11x2-coshaxdx – what'sup Oct 18 '13 at 13:56
  • This is 3.522.4 from Gradshtein & Ryzhik. – user64494 Oct 18 '13 at 14:10
  • @DavidH Actually，it's a physical problem which can be found in the book$$ Beautiful models :70 years of exactly solved quantum many-body problems $$. Unfortunately, I can't remember the exact page,but in the chapter,somewhat Ising-Heisenberg model. – Roger209 Oct 19 '13 at 03:09

2 Answers2

1

i think the OP wants to evaluate it

By using Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$

$$ I = \int_0^{\infty} \frac{dx}{\cosh(ax)(1+x^2)} $$

$$ u = bx \Rightarrow I = \frac{1}{b}\int_0^{\infty} \frac{dx}{\cosh \left(\frac{a}{b}x \right)\left( 1 + \frac{x^2}{b^2} \right)} $$

put $ \frac{a}{b} = \pi $

Community
  • 1
what'sup
  • 1,760
0

The integrand is even. Switch the integral to $\left(-\infty,\infty\right)$. Use complex integration $\left(~\mbox{Residue}\ \mbox{Theorem}~\right)$. Poles of $\cosh\left(\pi x\right)$ are at ${\rm i}\left(n + 1/2\right)$. $n \in {\mathbb Z}$. The "other piece" has poles at $z = \pm{\rm i}a^{-1/2}$. That's all we need.

Felix Marin
  • 89,464