CVR7
From Exampleproblems
Find the residues of 
at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where is given by
This has poles at 
of multiplicity 
.
Use the formula ![\mathrm{Res}_{z=z_0} f(z) = \frac{1}{(k-1)!}\lim_{z\to z_0} \frac{d^{k-1}}{{dz}^{k-1}}\left[(z-z_0)^k f(z)\right]\,](/wiki/images/math/f/d/e/fded86bdbecf9ea9fc9c85714763d2fa.png)
![\mathrm{Res}_{z=n\pi} =\lim_{z\to n\pi} \frac{d}{dz} (z-n\pi)^2 f(z) = \lim_{z\to n\pi} \frac{d}{dz} \left[ \frac{(z-n\pi)\cos z}{\sin z} \right]^2\,](/wiki/images/math/c/4/5/c45834661100498891fe4913aa6b766b.png)
![=\lim_{z\to n\pi} 2 \left[\frac{(z-n\pi)\cos z}{\sin z}\right] \left[ \frac{\sin z \left[(z-n\pi)(-\sin z) + \cos z \right] - (z-n\pi) \cos^2 z}{\sin^2(z)}\right] =0 \,](/wiki/images/math/e/5/5/e55cb6e69d4a4224fc64995ad34b3ffa.png)
In this case infinity is a limit point of singular points, so there is no residue at infinity.
