# Residue Theorem

1. Mar 3, 2009

### SNOOTCHIEBOOCHEE

1. The problem statement, all variables and given/known data

If f1 and f2 have residues r1 and r2 at z0. show that the residue of f1+ f2 is r1 + r2

3. The attempt at a solution

Res(f1, z0) = limz-->z0 (z-z0)f1(z) = r1
Res(f2, z0) = limz-->z0 (z-z0)f2(z) = r2

now calculate Res(f1+f2, z0)

=

limz-->z0 (z-z0)(f1(z)+ f2(z))
= limz-->z0 (z-z0)(f1(z))+(z-z0)(f2(z) = r1+ r2

Is it really this easy? i must be doing something wrong
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 3, 2009

### HallsofIvy

Staff Emeritus
Pretty much, yeah. "Residue" is only defined for poles and a function has a pole at $z_0$ if and only if it can be expanded in a power series (a Laurent series) with a finite number of negative exponents. In that case the residue is the coefficient of z-1 (thus that limit formula you use). Adding the two functions, you can add the Laurent series term by term: $a_1z^{-1}+ a_2z^{-2}= (a_1+ a_2)z^{-1}$. The residues add.

3. Mar 3, 2009

### SNOOTCHIEBOOCHEE

Thanks halls.