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Many titles on advanced topics are also available, such as complex variables and topology, but these may be harder to find in retail stores. Schaum's Outlines are frequently seen alongside the Barron's "Easy Way" series and McGraw-Hill's own "Demystified" series. The "Demystified" series is introductory in nature, for middle and high school ...
As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z i. Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.
An Introduction to Complex Analysis in Several Variables. Van Nostrand. Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw-Hill. ISBN 9780070542358. Rudin, Walter (1986). Real and Complex Analysis (International Series in Pure and Applied Mathematics). McGraw-Hill.
Complex analysis is the branch of mathematics investigating holomorphic functions, i.e. functions which are defined in some region of the complex plane, take complex values, and are differentiable as complex functions. Complex differentiability has much stronger consequences than usual (real) differentiability.
Schaum's Outline of Advanced Calculus (1963) [2010] Schaum's Outline of Complex Variables (1964) [2009] Schaum's Outline of Laplace Transforms (1965) Schaum's Mathematical Handbook of Formulas and Tables (1968) [2008] Schaum's Outline of Vector Analysis [And An Introduction to Tensor Analysis] (1968) [2009] Schaum's Outline of Real Variables (1969)
Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation. The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around C of every other term not in the form cz −1 is zero, and the integral is ...
Morera's theorem states that a continuous, complex-valued function f defined on an open set D in the complex plane that satisfies = for every closed piecewise C 1 curve in D must be holomorphic on D. The assumption of Morera's theorem is equivalent to f having an antiderivative on D .
In complex analysis, the open mapping theorem states that if is a domain of the complex plane and : is a non-constant holomorphic function, then is an open map (i.e. it sends open subsets of to open subsets of , and we have invariance of domain.).