Mittag-Leffler Cotangent expansion proof Complex analysis
Hello! I have this problem for my assignment, and I'm stuck at step 2, any help would be greatly appreciated thanks!
Prove the formula claimed in the
lecture as follows:
1)We know, looking at residues, that the difference:
is an entire function (that is, it is holomorphic everywhere on C. Compute the derivative of this difference (using cot = -1/sin^2) by reordering the sum.
2)Prove that for |Im(z)|-> inf, d'(z) -> 0. By the previous step you can assume w.l.o.g. that 0 < Re(z) <= 1! (Consider
the term involving sin and the infinite sum separately.)
I can solve the rest of the problem but I can't seem to prove that the lim as the imaginary part of zgoes to infinity makes d'(z) -> 0
The derivative I get is:
d'(z)= -pi^2/ sin^2(pi*z) +1/z^2+ pi/z *coth(z*pi)