Complex analysis (also known as the theory of functions of a complex variable) is one of the most useful branches of pure mathematics which is used to investigate the functions of complex numbers. It is a fundamental requirement of advanced mathematics and various disciplines of sciences including applied mathematics, algebra, number theory, physics, electrical and electronics engineering, etc. Complex analysis is concerned with analytic functions of complex variables or meromorphic functions and is used widely to solve two dimensional problems in mathematics and physics.
It is a new branch of mathematics than others and has its roots in earlier 19th century. Euler, Gauss, Riemann, Cauchy, Weierstrass and several other prominent scholars have developed complex analysis with their efforts and made it useful and beneficial for other disciplines of mathematics and sciences.
Complex analysis is a vast subject and has its own theorems and functions to solve the equation including Holomorphic function, Antiholomorphic function, Cauchy-Riemann equations, Conformal mapping, Power series, Radius of convergence, Laurent series, Meromorphic function, Entire function, Pole (complex analysis), Zero (complex analysis), Residue (complex analysis), singularity (Isolated, removable and essential), Branch point, Principal branch, Weierstrass-Casorati theorem, Landau’s constants, Holomorphic functions are analytic, Schwarzian derivative, Analytic capacity, Disk algebra, Bieberbach conjecture, Borel-Carathéodory theorem, Hadamard three-circle theorem, Hardy theorem, progressive function and Corona theorem, Nevanlinna theory, Picard’s theorem, Paley-Wiener theorem, Value distribution theory of holomorphic functions, Contour integrals, Line integral, Cauchy integral theorem, Cauchy’s integral formula, Residue theorem, Liouville’s theorem (complex analysis), Examples of contour integration, Fundamental theorem of algebra, Simply connected, Winding number, Morera’s theorem, Mellin transform, Kramers–Kronig relation, Analytic continuation, Antiderivative (complex analysis), Bôcher’s theorem, Carathéodory’s theorem (conformal mapping), Cayley transform, Complex differential equation, Harmonic conjugate, Hilbert’s inequality, Method of steepest descent, Mittag-Leffler’s theorem, Montel’s theorem, Periodic points of complex quadratic mappings, Pick matrix, Riemann mapping theorem, Riemann sphere, Riemann surface, Riemann-Roch theorem, Runge approximation theorem, Schwarz lemma and Weierstrass factorization theorem.