Q16P - Mathematical Methods in Physical Sciences

Question

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

Step-by-Step Solution

Answer

Sum of the residues at the singularity is zero.

Step 1: 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:

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

Step 2: 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.