Caratheodory C. Theory of Functions of a Complex Variable. Vol. 2

Chelsea Publishing Company, 1954. — 220 p.The present volume contains Parts Six and Seven of the entire work. The first of these two parts is devoted to the foundations of Geometric Function Theory, its three chapters dealing with bounded functions and with conformal mapping. Among the topics treated here, we mention G. Pick's beautiful interpretation of Schwarz's Lemma and a rather detailed account of the theory of the angular derivative. For the important theorem of Fatou on the boundary values of bounded functions, Caratheodory has chosen his own proof, which dates back to the year 1912.
The four chapter of the seventh and last part deal with the triangle functions and Picard's theorems. There are tables summarizing all of the fundamental solutions of this equation and giving the exceptional cases as well as the connecting formulas. This provides the student with all of the analytic tools required for the mapping of circular-arc triangles. The third chapter gives a geometric study of certain special cases, namely of the Schwarz triangle nets and the modular configuration. The fourth chapter gives the most important theorems on the exceptional values of meromorphic functions.