. The aim of this problem is to describe the automorphism group of an annulus, and in the process also answer the following question: when are two annuli conformally equivalent? The modulus of an annulus {z : a < |z − z0| < b} with inner radius a and outer radius b is defined to be 1 2π log b a . (a) Show that any conformal map from one annulus centred at the origin to another such annulus extends to a confor