Functions of a Complex Variable II contains the applications of the theory of differentiation and integration of complex functions begun in Volume I. 

First there is a discussion of conformal mappings and harmonic functions. 

An account of Laurent's Theorem and singularities leads to the calculation of integrals by residues and the book closes with a description of analytic continuation and Riemann surfaces. 

As in Volume I there are many worked examples and diagrams to illustrate the basic ideas.