Abstract
AbstractFunctions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. Multivalued functions can be made single-valued on a multi-sheet Riemann surface. The values of an analytic function in a region of the complex plane are completely defined by the knowledge of their values on a closed boundary of the region. Important properties and techniques of complex analysis are described. These include Taylor and Laurent expansions, contour integration and residue calculus, Green's functions, Laplace transforms and Bromwich integrals, dispersion relations and asymptotic expansions. Analytic functions are defined by their properties at the locations called singularities (poles and branch cuts) where they cease to be analytic. This feature makes analytic functions of particular interest in the construction of physical theories.
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