Neha Patil (Editor)

List of complex analysis topics

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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates 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.

Contents

Overview

  • Complex numbers
  • Complex plane
  • Complex functions
  • Complex derivative
  • Holomorphic functions
  • Harmonic functions
  • Elementary functions
  • Polynomial functions
  • Exponential functions
  • Trigonometric functions
  • Logarithmic functions
  • Inverse trigonometric functions
  • Residue theory
  • Isometries in the complex plane
  • 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

    • 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-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

  • Antiderivative (complex analysis)
  • Bôcher's theorem
  • Cayley transform
  • Complex differential equation
  • 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

  • Analytization trick
  • Biholomorphy
  • Cartan's theorems A and B
  • Cousin problems
  • Edge-of-the-wedge theorem
  • Several complex variables

    • People

  • Augustin Louis Cauchy
  • Leonhard Euler
  • Carl Friedrich Gauss
  • Jacques Hadamard
  • Kiyoshi Oka
  • Bernhard Riemann
  • Karl Weierstrass

    • References

    List of complex analysis topics Wikipedia
    (Text) CC BY-SA


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