Suppose that a function f is analytic at z0, and write g(z) = f (z)/(z z0).

Question:

Suppose that a function f is analytic at z0, and write g(z) = f (z)/(z − z0). Show that
(a) If f(z0) ≠ 0, then z0 is a simple pole of g, with residue f (z0);
(b) If f(z0) = 0, then z0 is a removable singular point of g.
Suggestion: As pointed out in Sec. 57, there is a Taylor series for f (z) about z0 since f is analytic there. Start each part of this exercise by writing out a few terms of that series.