In this reprint of his 1973 classic, Ahlfors explains conformal invariants and extremal problems, emphasizing the geometric approach as well as classical and semi-classical results that the late Finnish mathematician felt every student of complex analysis should know before starting independent research. This volume is the first time that most of its contents appeared in book form, including the author's discussion of the theory of extremal length and Schiffer's variational method, and a proof of $\vert a_4\vert \leq 4$. The book also introduces Riemann surfaces, with topological and analytical background that support a proof of the uniformization theorem. An extensive errata is also provided. Annotation ©2011 Book News, Inc., Portland, OR (booknews.com)