Show Res(F'(z)/F(z), z0) = m if F(z) is analytic on the disc |z - z0| < R and has a zero of order m at z0.
The Attempt at a Solution
We know that the kth derivation of F(z) is 0 for all k less than m, since F(z) has a zero of order m at z0. The kth derivative of F(z) is not 0 for all k greater than or equal to m.
We know F(z) can be written as a power series on the disc and that the coefficients are given by the kth derivative of F(z), evaluated at z0 and divided by k!. That means that the power series for F(z) = ∑(m to ∞) an(z-z0)^n since an = 0 for all n less than m.
I want to show F'(z)/F(z) has a residue at z0 which is m, but am not quite sure how I should begin. I tried solving by equating coefficients for some function G(z), where I showed F(z)G(z) = F'(z) but this seems wrong.