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- Homework Statement
- Find ##Res(h,z_0)##

- Relevant Equations
- ##h=\frac{f}{g}##

There is a typo. It should say ##h=\frac{f}{g}##.

Attempt: ##f## and ##g## are holomorphic on ##\Omega##. Homomorphic functions form a ##\mathcal{C}^*## algebra, so ##h## is holomorphic on ##\Omega## where ##g\neq 0##.

If ##z_0## is a removal singularity of ##h##, then ##Res(h,z_0)=0## by Morera's theorem.

Assume ##f(z_0)\neq 0##, then ##z_0## is a second order pole of h since ##g''(x_0)\neq 0 ##.

$$Res(h,z_0)=lim_{z\rightarrow z_0}\frac{d}{dz}\Big[\frac{f}{g}(z-z_0)^2\Big]$$

This method produces indeterminate forms, even after applying the L'Hopital's rule.

Let ##\mathcal{C}=\{z:|z-z_0|=1/2\}##. By Residue theorem,

$$Res(h,z_0)=\frac{1}{2\pi i}\int_\mathcal{C}\frac{f}{g}dz$$

This integral cannot be computed since ##f## and ##g## are not given.

Can anyone think of a clever method that could solve this problem?

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