Consider the function

Where q is analytic at z0, q(z0) = 0, and q'(z0) ‰  0. Show that z0 is a pole of order m = 2 of the function f, with residue

Suggestion: that z0 is a zero of order m = 1 of the function
q, so that
q(z) = (z ˆ’ z0)g(z)
Where g(z) is analytic and nonzero at z0. Then write

The desired form of the residue B0 = φ' (z0) can be obtained by showing that
q'(z0) = (z0) and q'' (z0) = 2g' (z0).

f(z) [a(2)12 q" (z0) (z) f(z)--(2) g(2)] 2