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Augustin-Louis Cauchy, one of the founders of complex analysis. Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Gösta Mittag-Leffler, Weierstrass, and many more in the 20th century.
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity ...
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 Invitation to Analytic Combinatorics: From One to Several Variables (PDF). Springer Texts & Monographs in Symbolic Computation. Pemantle, Robin; Wilson, Mark C. (2013). Analytic Combinatorics in Several Variables (PDF). Cambridge University Press. Sedgewick, Robert. "4. Complex Analysis, Rational and Meromorphic Asymptotics" (PDF)
The maximum modulus principle has many uses in complex analysis, and may be used to prove the following: The fundamental theorem of algebra. Schwarz's lemma, a result which in turn has many generalisations and applications in complex analysis. The Phragmén–Lindelöf principle, an extension to unbounded domains.
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon.Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction.
Complex Analysis treats the standard topics of a course in complex variables as well as several applications to other areas of mathematics. [ 2 ] [ 10 ] The chapters cover the complex plane , Cauchy's integral theorem , meromorphic functions , connections to Fourier analysis, entire functions , the gamma function , the Riemann zeta function ...
In mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial. It is named for Jacques Hadamard.