Volume III sets out classical Cauchy theory. It is much more geared towards its innumerable
applications than towards a more or less complete theory of analytic functions. Cauchy-type
curvilinear integrals are then shown to generalize to any number of real variables
(differential forms Stokes-type formulas). The fundamentals of the theory of manifolds are
then presented mainly to provide the reader with a canonical'' language and with some
important theorems (change of variables in integration differential equations). A final
chapter shows how these theorems can be used to construct the compact Riemann surface of an
algebraic function a subject that is rarely addressed in the general literature though it only
requires elementary techniques. Besides the Lebesgue integral Volume IV will set out a piece
of specialized mathematics towards which the entire content of the previous volumes will
converge: Jacobi Riemann Dedekind series and infinite products elliptic functions classical
theory of modular functions and its modern version using the structure of the Lie algebra of
SL(2 R).
