1. ## rectangular contours

Can someone help me with this question.
Evaluate the integral fro 0 to inifinity of sinx over sinh x ?

Thanks

2. Originally Posted by jules
Can someone help me with this question.
Evaluate the integral fro 0 to inifinity of sinx over sinh x ?

Thanks
$\int_0^{\infty} \frac{\sin x}{\sinh x}dx$.
$\int_{-\infty}^{\infty} \frac{\sin x}{\sinh x} dx$
Because the function is even.
Now define the complex-valued function,
$f(z) = \frac{\sin z}{\sinh z}$.
The singularities of this function occur when $\sinh z = 0$. Since there are infinitely many such values we cannot use the semi-circular contour. Let us use a rectangular contour: $-R,+R,-R+\frac{3}{2}\pi i,R+\frac{3}{2}\pi i$.
Can you finish?

3. ## rectangular contours

Thanks that really got me started. Though im stuck again im trying to compute the residue. Can you help please

4. Originally Posted by jules
Thanks that really got me started. Though im stuck again im trying to compute the residue. Can you help please
Can you post the full integral (including all the 4 sides) so I can see what you did.

Do you understand why I chose such a contour?