Forsyth A.R. Theory of functions of a complex variable

Cambridge University Press, 1893. — 704 p.This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible.
The book may be considered as composed of five parts.
The first part, consisting of Chapters I—VII, contains the theory of uniform functions : the discussion is based upon powerseries, initially connected with Cauchy's theorems in integration, and the properties established are chiefly those which are contained in the memoirs of Weierstrass and Mittag-Leffler.
The second part, consisting of Chapters VIII—XIII, contains the theory of multiform functions, and of uniform periodic functions which are derived through the inversion of integrals of algebraic functions. The method adopted in this part is Cauchy's, as used by Briot and Bouquet in their three memoirs and in their treatise on elliptic functions : it is the method that has been followed by Hermite and others to obtain the properties of various kinds of periodic functions. A chapter has been devoted to the proof of Weierstrass's results relating to functions that possess an addition-theorem.
The third part, consisting of Chapters XIV—XVIII, contains the development of the theory of functions according to the method initiated by Biemann in his memoirs. The proof which is given of the existence-theorem is substantially due to Schwarz; in the rest of this part of the book, I have derived great assistance from Neumann's treatise on Abelian functions, from Fricke's treatise on Klein's theory of modular functions, and from many memoirs by Klein.
The fourth part, consisting of Chapters XIX and XX, treats of conformal representation. The fundamental theorem, as to the possibility of the conformal representation of surfaces upon one another, is derived from the existence-theorem : it is a curious fact that the actual solution, which has been proved to exist in general, has been obtained only for cases in which there is distinct limitation.
The fifth part, consisting of Chapters XXI and XXII, contains an introduction to the theory of Fuchsian or automorphic functions, based upon the researches of Poincare and Klein : the discussion is restricted to the elements of this newly-developed theory.