# Thread: Mittag-Leffler Cotangent expansion proof Complex analysis

1. ## 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!!

2. Originally Posted by Jimena
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)
I think you would be better off leaving the derivative in the form $\displaystyle d'(z) = -\frac{\pi^2}{\sin^2\pi z} + \frac1{z^2} + \sum_{k\in\mathbb{Z}\setminus\{0\}}\frac1{(z-k)^2}.$

In the sum, group the terms with index k and –k (that's what the hint about reordering the sum is getting at!), to make it $\displaystyle \sum_{k=1}^\infty \frac{2(z^2+k^2)}{(z^2-k^2)^2}$. In that form, it's easier to see what happens as |Im(z)| goes to infinity.

This is all spelt out in Ahlfors' Complex Analysis (Section 5.2 in the edition that I have). If you want to get to grips with complex analysis, Ahlfors is really an essential possession.