I checked your calculations and they seem correct, but I'm not convinced by the theorem.
Are you sure this holds for arbitrary f(z)? Perhaps you could state the theorem, with the conditions.
According to a theorem I have seen in class, we have
It can easily be seen that the theorem works for , but I've been trying to use it to find the exact value of without success. The poles of fare obviously i and -i, for which the residues are respectively and , which implies that
Of course, f is an even function, so .
But, when I approximate the sum in Maple or MATLAB, I get another answer (which is the right one): -0.3639849730...
Does anyone have an idea of what I have done wrong?
Thank you all!!
The only conditions I'm aware of if that must be analytic , and that when .
I still don't understand why this doesn't work, but I've just learned a new theorem which says, under the same conditions, that if we have to evaluate in the series, the same formula works except that we must use instead of to calculate the residue. This last formulae works for the series I wanted to evaluate initially and gives a value of .
Thank you for checking my calculations... I'm still curious about why the first theorem didn't work. Maybe I'll ask my teacher about that...