Question

To prove that the sum of the residues at finite points plus the residence at infinity is zero.

Short answer

Sum of the residues at the singularity is zero.

Step-by-step solution

Use residue theorem

Proof: - Let C be a closed contour which encloses all the singularities of f(z) except that at infinity, then by residue theorem as follows:

${\int }_{c}f\left(z\right)dz=2\mathrm{\pi i}\sum _{}^{}\mathrm{R}+\Rightarrow \sum _{}\mathrm{R}+=\underset{2\mathrm{\pi i}}{1}{\int }_{\mathrm{c}}\mathrm{f}\left(\mathrm{z}\right)\mathrm{dz}$

By definition we know that:

$-\frac{1}{2\mathrm{\pi i}}{\int }_{\mathrm{c}}\mathrm{f}\left(\mathrm{z}\right)\mathrm{dz}=\mathrm{Res}(\mathrm{z}=\infty )$

Add the above equations for the solution

By adding, obtain:

$\sum _{}^{}{R}^{+}Res(z=\infty )=0$

That is, sum of the residues at the singularity is zero.