MHB Compute Residues for Periodic Function with Multiple Poles?

[pantboio]

How can i compute $Res(f,z_k)$ where
$$f(z)=\frac{z-1}{1+cos\pi z}$$
and $z_k=2k+1, k\neq 0$?

 

[Ackbach]

I think it's not too difficult to show that the $z_{k}$'s are simple poles. [EDIT] This is incorrect. $z_{0}$ is simple, but the others are double.

 

[Ackbach]

$z_k$ are zeros of order two of denominator, but $z_0=0$ is also a zero of numerator. So $z_0=0$ is a simple pole, but all the others $z_k=2k+1$ with $k\neq 0$ are 2-poles.

Whoops. You're right. So you're going to need the next order formula, assuming that works. You could try
$$ \text{Res}(f,z_{k})= \lim_{z \to z_{k}} \frac{d}{dz} \left[(z-z_{k})^{2}f(z) \right]$$
for the second-order poles. That would come out to
$$ \text{Res}(f,2k+1)= \lim_{z \to 2k+1} \frac{d}{dz} \left[(z-2k-1)^{2} \, \frac{z-1}{1+ \cos( \pi z)} \right]$$
$$= \lim_{z \to 2k+1} \frac{d}{dz} \left[ \frac{(z-1)(z-2k-1)^{2}}{1+ \cos( \pi z)} \right]$$
$$= \lim_{z \to 2k+1} \frac{d}{dz} \left[ \frac{z^{3}-(4k+3)z^{2} +(4k^{2}+8k+3) z-(2k+1)^{2}}{1+ \cos( \pi z)} \right].$$
Looks like a fair amount of algebra. Hmm. You might be better off going for the series expansion.

 

[Fernando Revilla]

Using a series expansion: $1+\cos \pi z=\ldots=\dfrac{\pi^2}{2}[z-(2k+1)]^2+\ldots$. Now,

$\dfrac{z-1}{1+\cos \pi z}=\dfrac{2k+[z-(2k+1)]}{\dfrac{\pi^2}{2}[z-(2k+1)]^2+\ldots}=\ldots +\dfrac{A_{-2}}{[z-(2k+1)]^2}+\dfrac{2/\pi^2}{z-(2k+1)}+A_0+\ldots$

So, $\mbox{Res }(f,2k+1)=\mbox{coef }\left(\dfrac{1}{z-(2k+1)}\right)=\dfrac{2}{\pi^2}$

 

[pantboio]

Using a series expansion: $1+\cos \pi z=\ldots=\dfrac{\pi^2}{2}[z-(2k+1)]^2+\ldots$. Now,

$\dfrac{z-1}{1+\cos \pi z}=\dfrac{2k+[z-(2k+1)]}{\dfrac{\pi^2}{2}[z-(2k+1)]^2+\ldots}=\ldots +\dfrac{A_{-2}}{[z-(2k+1)]^2}+\dfrac{2/\pi^2}{z-(2k+1)}+A_0+\ldots$

So, $\mbox{Res }(f,2k+1)=\mbox{coef }\left(\dfrac{1}{z-(2k+1)}\right)=\dfrac{2}{\pi^2}$

so the residue is the same for all poles, maybe because of $\cos$ periodicity

 

[Fernando Revilla]

so the residue is the same for all poles, maybe because of $\cos$ periodicity

More general: the coefficient $A_{-1}$ in the Laurent series expansion of $f$ does not depend on $k$.