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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)

Thank you!!