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**transgalactic** $\displaystyle f=\frac{z}{1-\cos z}$

the singular points are z=2pik

and poles because there limit is infinity

now i want to determine te power of the pole

g=1/f=$\displaystyle \frac{1-\cos z}{z}$

$\displaystyle g'=\frac{(-\sin z)z-(1-\cos z)}{z^2}$

$\displaystyle g'(2\pi k)=0$

$\displaystyle g''=\frac{-\sin z z^2 -(cos z -1)2z}{z^4}$

$\displaystyle g''(2\pi k)=0$

the book says that its a second order pole

which is not true

because $\displaystyle g''(2\pi k)=0$

where it should differ zero in order to be pole