I know answer for $$ \oint_{|z-\frac{\pi}{2}(1-i)|= \pi} \frac{z\,dz}{\cos z-\cosh z} = 2\pi i(1-e^{-1})$$ But I don't understand why it's true.
I know that $$ \oint_{|z-\frac{\pi}{2}(1-i)|= \pi} \frac{z\,dz}{\cos(z)-\cosh(z)} =\sum_{z_0 \in D} \mathrm{Rez}\,f(z)$$ where D - domain into contour and rez - deduction at a particular point. I find one point into domain $$z=0$$ and $$\mathrm{Rez}\,f(0) = -1 $$ But it's not correct answer. Can you help me find my error?
$\cosh z$ hyperbolic cosine function given by $$ \cosh z = \frac{e^{z}+e^{-z}}{2}$$