I was trying to find $$ \mathrm{Res}(f(z)=z^3 \cos(z^{-2}),z=0)$$ I tried to calculate it with the $\frac{1}{2\pi i}$ multipled with the integral of $f(z)$ on the circle $$|z|=1.$$
While solving I found my self having to solve the integral of $\exp(iz)$ and from here I couldn't continue..
2026-03-30 00:00:02.1774828802
What is the integral of $\exp(iz)$?
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Use the Taylor series of $\cos z$ to find the Laurent series of $f(z)$
$\cos z = 1 - \frac {z^2}{2} + \frac {z^4}{4} - \cdots\\ z^3\cos z^{-2} = z^3 - \frac {z^{-1}}{2} + \frac {z^{-5}}{4} - \cdots$
We only care about the coefficient of the $z^{-1}$ term
$\frac 12$