Residue Theorem: Show r1+r2 = Res(f1+f2, z0)

[SNOOTCHIEBOOCHEE]

Homework Statement

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

The Attempt at a Solution

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

now calculate Res(f₁+f₂, z₀)

=

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

Is it really this easy? i must be doing something wrong

 

[HallsofIvy]

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.

 

[SNOOTCHIEBOOCHEE]

Thanks halls.