Editing List of complex analysis topics

{{Short description|none}}
'''[[Complex analysis]]''', traditionally known as the '''theory of functions of a complex variable''', is the branch of [[mathematics]] that investigates [[Function (mathematics)|functions]] of [[complex numbers]]. It is useful in many branches of mathematics, including [[number theory]] and [[applied mathematics]]; as well as in [[physics]], including [[hydrodynamics]], [[thermodynamics]], and [[electrical engineering]].

See also: [[glossary of real and complex analysis]].

==Overview==
* [[Complex numbers]]
* [[Complex plane]]
* [[Complex functions]]
** [[Complex derivative]]
** [[Holomorphic functions]]
** [[Harmonic functions]]
* [[Elementary functions]]
** [[Polynomial function]]s
** [[Exponential functions]]
** [[Trigonometric functions]]
**[[Hyperbolic functions]]
** [[Logarithmic functions]]
** [[Inverse trigonometric functions]]
**[[Inverse hyperbolic functions]]
* [[Residue theory]]
* [[Euclidean plane isometry#Isometries in the complex plane|Isometries in the complex plane]]

==Related fields==
{{Main|Applied mathematics}}

* [[Number theory]]
* [[Hydrodynamics]]
* [[Thermodynamics]]
* [[Electrical engineering]]

==Local theory==

*[[Holomorphic function]]
*[[Antiholomorphic function]]
*[[Cauchy–Riemann equations]]
*[[Conformal mapping]]
**[[Conformal welding]]
*[[Power series]]
*[[Radius of convergence]]
*[[Laurent series]]
*[[Meromorphic function]]
*[[Entire function]]
*[[Pole (complex analysis)]]
*[[Zero (complex analysis)]]
*[[Residue (complex analysis)]]
*[[Isolated singularity]]
*[[Removable singularity]]
*[[Essential singularity]]
*[[Branch point]]
*[[Principal branch]]
*[[Weierstrass–Casorati theorem]]
*[[Landau's constants]]
*[[Holomorphic functions are analytic]]
*[[Schwarzian derivative]]
*[[Analytic capacity]]
*[[Disk algebra]]
*[[Univalent function]]

==Growth and distribution of values==

*[[Ahlfors theory]]
*[[Bieberbach conjecture]]
*[[Borel–Carathéodory theorem]]
*[[Corona theorem]]
*[[Hadamard three-circle theorem]]
*[[Hardy space]]
*[[Hardy's theorem]]
*[[Maximum modulus principle]]
*[[Nevanlinna theory]]
*[[Paley–Wiener theorem]]
*[[Progressive function]]
*[[Value distribution theory of holomorphic functions]]

==Contour integrals==

*[[Line integral]]
*[[Cauchy's 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]]
**[[Principle of the argument]]
**[[Rouché's theorem]]
*[[Bromwich integral]]
*[[Morera's theorem]]
*[[Mellin transform]]
*[[Kramers–Kronig relation]], a. k. a. [[Hilbert transform]]
*[[Sokhotski–Plemelj theorem]]

==Special functions==

*[[Exponential function]]
*[[Beta function]]
*[[Gamma function]]
*[[Riemann zeta function]]
**[[Riemann hypothesis]]
**[[Generalized Riemann hypothesis]]
*[[Elliptic function]]
**[[Half-period ratio]]
**[[Jacobi's elliptic functions]]
**[[Weierstrass's elliptic functions]]
**[[Theta function]]
*[[Elliptic modular function]]
*[[j-invariant|J-function]]
*[[Modular function]]
*[[Modular form]]

==Riemann surfaces==

*[[Analytic continuation]]
*[[Riemann sphere]]
*[[Riemann surface]]
*[[Riemann mapping theorem]]
*[[Carathéodory's theorem (conformal mapping)]]
*[[Riemann–Roch theorem]]

==Other==

*[[Amplitwist]]
*[[Antiderivative (complex analysis)]]
*[[Bôcher's theorem]]
*[[Cayley transform]]
*[[Harmonic conjugate]]
*[[Hilbert's inequality]]
*[[Method of steepest descent]]
*[[Montel's theorem]]
*[[Periodic points of complex quadratic mappings]]
*[[Pick matrix]]
*[[Runge approximation theorem]]
*[[Schwarz lemma]]
*[[Weierstrass factorization theorem]]
*[[Mittag-Leffler's theorem]]
*[[Sendov's conjecture]]
* [[Infinite compositions of analytic functions]]

==Several complex variables==
*[[Biholomorphy]]
*[[Cartan's theorems A and B]]
*[[Cousin problems]]
*[[Edge-of-the-wedge theorem]]
*[[Several complex variables]] <!-- please do not link to the section above, that's bad style, see the [[WP:MoS]] -->

==History==
{{Main|History of complex analysis}}

==People==

* [[Augustin Louis Cauchy]]
* [[Leonhard Euler]]
* [[Carl Friedrich Gauss]]
* [[Jacques Hadamard]]
* [[Kiyoshi Oka]]
* [[Bernhard Riemann]]
* [[Karl Weierstrass]]
* [[Pierre Alphonse Laurent]]
* [[Brook Taylor]]
* [[Siméon Denis Poisson]]
* [[Hermann Schwarz]]
* [[Camille Jordan]]
* [[Carl Gustav Jacob Jacobi]]
* [[Eugène Rouché]]
* [[Gerardus Mercator]]
* [[Joseph Liouville]]
* [[Pierre-Simon Laplace]]
* [[August Ferdinand Möbius]]
* [[William Kingdon Clifford]]

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