Branches of Mathematic

Complex Analysis

Analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that studies complex valued functions of one or more complex variables. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Central Question
Which functions of a complex variable are holomorphic, differentiable in the complex sense, and what unusual rigidity does that single condition force on such a function's behavior. 1
Key Debate
Bernhard Riemann's use of the Dirichlet principle to prove foundational results in complex analysis was challenged when Karl Weierstrass published a 1870 counterexample showing the principle's underlying assumption could fail, undermining Riemann's reasoning until David Hilbert rehabilitated the method in 1900. 2
Classification
Pure or Applied
Both / Interdisciplinary 1
Connections

Associated With

Includes

Abel's Test, Theorems
Source Abel's test, Wikipedia
Source Adolf Hurwitz (MacTutor)
Source Adolph Winkler Goodman (Wikipedia)
Source Alfred Enneper (Wikipedia)
Source Alfred Tauber (MacTutor)
Source Allen Shields (Wikipedia)
Source Complex Analysis (Wikipedia)
Source Wolfram MathWorld: Barnes G-Function
Source Complex Analysis (Wikipedia)
Source Bloch's theorem (complex analysis) (Wikipedia)
Source Borel-Caratheodory theorem - Wikipedia
Source Brennan Conjecture (Wikipedia)
Source Carl Johan Malmsten (Wikipedia)
Source Carl Johannes Thomae (MacTutor)
Source Casorati-Weierstrass theorem (Wikipedia)
Source Caspar Wessel (Wikipedia)
Source Charles-François-Maximilien Marie (Wikipedia)
Source Complex Number (Wikipedia)
Source Wolfram MathWorld: Complex Plane
Source Wikipedia: Constantin Carathéodory
Source Denjoy-Wolff theorem (Wikipedia)
Source Dimitrie Pompeiu (Wikipedia)
Source Donald C. Spencer (Wikipedia)
Source Edmond Laguerre (Wikipedia)
Source Edmund Landau, Biography (MacTutor History of Mathematics)
Source MacTutor biography of Édouard Goursat, St Andrews
Source Edward Burr Van Vleck (MacTutor)
Source Ernesto Pascal (Wikipedia)
Source Ernst Leonard Lindelöf (Wikipedia)
Source Eugène Rouché (Wikipedia)
Source Eugenio Elia Levi (Wikipedia)
Source Felice Casorati (MacTutor)
Source MacTutor biography of Felix Klein, St Andrews
Source Gábor Szegő biography (MacTutor)
Source Gauss-Lucas theorem (Wikipedia)
Source George Pólya (Wikipedia)
Source Giacinto Morera (Wikipedia)
Source Goodman's Conjecture (Wikipedia)
Source Wikipedia: Gösta Mittag-Leffler
Source Hadamard three-lines theorem, Wikipedia
Source Harry Rauch (Wikipedia)
Source Herbert Grötzsch (Wikipedia)
Source Hermann Hankel, Biography (MacTutor History of Mathematics)
Source Hermann Schwarz (Wikipedia)
Source Theta function (Wikipedia)
Source Joseph Liouville (Wikipedia)
Source Joukowsky transform (Wikipedia)

Showing 1 to 50 of 78 in Includes, page 1 of 2.

Sources
1. Complex Analysis (Wikipedia)
Wikipedia
  • lead paragraph
    Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that studies complex-valued functions of one or more complex variables.
  • Holomorphic functions section
    Complex analysis primarily studies holomorphic functions, which are differentiable functions of a complex variable.
  • Lead section, pure or applied classification
    It also has applications in engineering fields, such as nuclear, aerospace, mechanical, and electrical engineering.
  • Includes: Bernhard Riemann, History section
    Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century.
  • Includes: Augustin-Louis Cauchy, History section
    Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century.
View the Source
2. Dirichlet Principle (Wikipedia)
WikipediaHistory section
Quote, History section
Karl Weierstrass published the first criticism of this assumption in 1870, giving an example of a functional that has a greatest lower bound which is not a minimum value.
View the Source
Karl Weierstrass, Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsIncludes: Karl Weierstrass, Opening summary
Quote, Includes: Karl Weierstrass, Opening summary
Karl Weierstrass is best known for his construction of the theory of complex functions by means of power series.
View the Source
Complex Number (Wikipedia)
Wikimedia FoundationIncludes: Complex Number, Complex analysis section
Quote, Includes: Complex Number, Complex analysis section
The study of functions of a complex variable is known as complex analysis
View the Source
Riemann Sphere (Wikipedia)
Wikimedia FoundationIncludes: Riemann SphereView the Source
Theta function (Wikipedia)
Includes: Jacobi Theta Function, Lead
Quote, Includes: Jacobi Theta Function, Lead
theta functions are special functions of several complex variables
View the Source
Modular Lambda Function (Wikipedia)
Includes: Modular lambda function, Lead
Quote, Includes: Modular lambda function, Lead
is a highly symmetric holomorphic function on the complex upper half-plane
View the Source
Adolf Hurwitz (MacTutor)
Includes: Adolf Hurwitz, Biography, further topics
Quote, Includes: Adolf Hurwitz, Biography, further topics
Further topics studied by Hurwitz include complex function theory
View the Source
Adolph Winkler Goodman (Wikipedia)
Includes: Adolph Winkler Goodman, Lead, univalent functions
Quote, Includes: Adolph Winkler Goodman, Lead, univalent functions
contributed to number theory, graph theory and to the theory of univalent functions
View the Source
Carl Johannes Thomae (MacTutor)
Includes: Carl Johannes Thomae, Summary
Quote, Includes: Carl Johannes Thomae, Summary
Carl Johannes Thomae was a German mathematician who worked in function theory.
View the Source
Edmond Laguerre (Wikipedia)
Includes: Edmond Laguerre, Lead, areas of work
Quote, Includes: Edmond Laguerre, Lead, areas of work
His main works were in the areas of geometry and complex analysis.
View the Source
Dimitrie Pompeiu (Wikipedia)
Includes: Dimitrie Pompeiu, Research
Quote, Includes: Dimitrie Pompeiu, Research
Pompeiu's contributions were mainly in the field of mathematical analysis, complex functions theory, and rational mechanics.
View the Source
Norman Levinson (Wikipedia)
Includes: Norman Levinson, Lead, fields of contribution (complex analysis)
Quote, Includes: Norman Levinson, Lead, fields of contribution (complex analysis)
Some of his major contributions were in the study of Fourier transforms, complex analysis, non-linear differential equations, number theory, and signal processing.
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Walter Rudin (Wikipedia)
Includes: Walter Rudin, Lead
Quote, Includes: Walter Rudin, Lead
In addition to his contributions to complex and harmonic analysis
View the Source
Giacinto Morera (Wikipedia)
Includes: Giacinto Morera, Lead
Quote, Includes: Giacinto Morera, Lead
He is known for Morera's theorem in the theory of functions of a complex variable
View the Source
Eugène Rouché (Wikipedia)
Includes: Eugène Rouché, Career
Quote, Includes: Eugène Rouché, Career
He is best known for Rouché's theorem in complex analysis
View the Source
Herbert Grötzsch (Wikipedia)
Includes: Herbert Grötzsch, Lead, field
Quote, Includes: Herbert Grötzsch, Lead, field
known primarily for his work on complex analysis
View the Source
Gábor Szegő biography (MacTutor)
Includes: Gábor Szegő, Biography, mathematical work
Quote, Includes: Gábor Szegő, Biography, mathematical work
worked mainly in function theory
View the Source
Harry Rauch (Wikipedia)
Includes: Harry Rauch, Lead
Quote, Includes: Harry Rauch, Lead
who worked on complex analysis and differential geometry
View the Source
Mstislav Keldysh (Wikipedia)
Includes: Mstislav Keldysh, Career, dissertation
Quote, Includes: Mstislav Keldysh, Career, dissertation
dissertation entitled Complex Variable and Harmonic Functions Representation by Polynomial Series
View the Source
Donald C. Spencer (Wikipedia)
Includes: Donald C. Spencer, Lead, several complex variables
Quote, Includes: Donald C. Spencer, Lead, several complex variables
and on several complex variables from the point of view of partial differential equations
View the Source
Lars Ahlfors (Wikipedia)
Includes: Lars Ahlfors, Lead
Quote, Includes: Lars Ahlfors, Lead
the leading figure in complex analysis during the 20th century
View the Source
Alfred Tauber (MacTutor)
Includes: Alfred Tauber, Summary, function theory
Quote, Includes: Alfred Tauber, Summary, function theory
known for his contribution to mathematical analysis and to the theory of functions of a complex variable
View the Source
Alfred Enneper (Wikipedia)
Includes: Alfred Enneper, Lead, dissertation
Quote, Includes: Alfred Enneper, Lead, dissertation
his dissertation about functions with complex arguments
View the Source
Carl Johan Malmsten (Wikipedia)
Includes: Carl Johan Malmsten, Lead, complex variable
Quote, Includes: Carl Johan Malmsten, Lead, complex variable
early research into the theory of functions of a complex variable
View the Source
Felice Casorati (MacTutor)
Includes: Felice Casorati (mathematician), Summary
Quote, Includes: Felice Casorati (mathematician), Summary
He also worked on complex function theory, and wrote an important book Teoria delle funzioni di variabili complesse
View the Source
Ernesto Pascal (Wikipedia)
Includes: Ernesto Pascal, Life and work
Quote, Includes: Ernesto Pascal, Life and work
Pascal was remembered for his work on elliptic functions based on Jacobi theta function.
View the Source
Edward Burr Van Vleck (MacTutor)
Includes: Edward Burr Van Vleck, Biography, research papers (function theory)
Quote, Includes: Edward Burr Van Vleck, Biography, research papers (function theory)
Almost all Van Vleck's research papers were in the fields of function theory and differential equations.
View the Source
Charles-François-Maximilien Marie (Wikipedia)
Includes: Charles-François-Maximilien Marie, Biography, 1862 appointment
Quote, Includes: Charles-François-Maximilien Marie, Biography, 1862 appointment
he wrote the book Théorie des fonctions de variables imaginaires relating to the Imaginary unit
View the Source
MacTutor History of Mathematics: Mikhail Alekseevich Lavrentev
MacTutor History of MathematicsIncludes: Mikhail Lavrentyev, Biography, Steklov chair
Quote, Includes: Mikhail Lavrentyev, Biography, Steklov chair
from 1933 he held the chair of the Theory of Functions of a Complex Variable at the Steklov Institute of Mathematics of the USSR Academy of Sciences
View the Source
Rolf Nevanlinna (Wikipedia)
Includes: Rolf Nevanlinna, Lead
Quote, Includes: Rolf Nevanlinna, Lead
made significant contributions to complex analysis
View the Source
Wilhelm Wirtinger (Wikipedia)
Includes: Wilhelm Wirtinger, Lead, complex analysis
Quote, Includes: Wilhelm Wirtinger, Lead, complex analysis
working in complex analysis, geometry, algebra, number theory, Lie groups, and knot theory
View the Source
Paul Montel (Wikipedia)
Includes: Paul Montel, Lead
Quote, Includes: Paul Montel, Lead
He researched mostly on holomorphic functions in complex analysis.
View the Source
Allen Shields (Wikipedia)
Includes: Allen Shields, Lead, complex analysis
Quote, Includes: Allen Shields, Lead, complex analysis
who worked on measure theory, complex analysis, functional analysis and operator theory
View the Source
Paul Julius Oswald Teichmüller (MacTutor)
Includes: Oswald Teichmüller, Biography, contributions
Quote, Includes: Oswald Teichmüller, Biography, contributions
He introduced quasi-conformal mappings and differential geometric methods into complex analysis.
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Caspar Wessel (Wikipedia)
Includes: Caspar Wessel, Lead
Quote, Includes: Caspar Wessel, Lead
geometrical interpretation of complex numbers
View the Source
Montel's theorem (Wikipedia)
Includes: Montel's Theorem, Lead sentence
Quote, Includes: Montel's Theorem, Lead sentence
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic function
View the Source
Hadamard three-lines theorem, Wikipedia
Includes: Hadamard Three-Lines Theorem, Lead sentence
Quote, Includes: Hadamard Three-Lines Theorem, Lead sentence
In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic funct
View the Source
Resolvent Formalism (Wikipedia)
Includes: Resolvent Formalism, Lead sentence
Quote, Includes: Resolvent Formalism, Lead sentence
rmalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and
View the Source
Casorati-Weierstrass theorem (Wikipedia)
Includes: Casorati-Weierstrass Theorem, Lead sentenceView the Source
Borel-Caratheodory theorem - Wikipedia
Includes: Borel-Caratheodory Theorem, Lead sentenceView the Source
Brennan Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Brennan Conjecture, Lead sentence
Quote, Includes: Brennan Conjecture, Lead sentence
In mathematics, specifically complex analysis, the Brennan conjecture is a conjecture estimating (under specified conditions) the
View the Source
Goodman's Conjecture (Wikipedia)
Includes: Goodman's Conjecture, Lead sentence
Quote, Includes: Goodman's Conjecture, Lead sentence
ficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.
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Wikipedia: Riemann surface
Includes: Riemann Surface, Lead sentence
Quote, Includes: Riemann Surface, Lead sentence
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold.
View the Source
Runge's Theorem (Wikipedia)
Wikimedia FoundationIncludes: Runge's Theorem, Lead sentence
Quote, Includes: Runge's Theorem, Lead sentence
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Ru
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Wikipedia: Mittag-Leffler's theorem
WikipediaIncludes: Mittag-Leffler Theorem, Lead sentence
Quote, Includes: Mittag-Leffler Theorem, Lead sentence
In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles.
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Morera's theorem, Wikipedia
Includes: Morera's Theorem, Lead sentence
Quote, Includes: Morera's Theorem, Lead sentence
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives a criterion for proving that a
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Phragmen-Lindelof principle (Wikipedia)
Includes: Phragmen-Lindelof Principle, Lead sentenceView the Source
Denjoy-Wolff theorem (Wikipedia)
Includes: Denjoy-Wolff Theorem, Lead sentenceView the Source
Mergelyan's theorem (Wikipedia)
Includes: Mergelyan's Theorem, Lead sentence
Quote, Includes: Mergelyan's Theorem, Lead sentence
is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
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Gauss-Lucas theorem (Wikipedia)
Includes: Gauss-Lucas Theorem, Lead sentenceView the Source
Bloch's theorem (complex analysis) (Wikipedia)
Includes: Bloch's Theorem (Complex Analysis), Lead sentence
Quote, Includes: Bloch's Theorem (Complex Analysis), Lead sentence
In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit
View the Source
Hermann Schwarz (Wikipedia)
Includes: Hermann Schwarz, Lead paragraph [in-branch]
Quote, Includes: Hermann Schwarz, Lead paragraph [in-branch]
complex analysis
View the Source
Joseph Liouville (Wikipedia)
Includes: Joseph Liouville, Lead paragraph [in-branch]
Quote, Includes: Joseph Liouville, Lead paragraph [in-branch]
complex analysis
View the Source
Siméon Denis Poisson (Wikipedia)
Includes: Siméon Denis Poisson, Lead paragraph [in-branch 2]
Quote, Includes: Siméon Denis Poisson, Lead paragraph [in-branch 2]
complex analysis
View the Source
Ernst Leonard Lindelöf (Wikipedia)
Includes: Ernst Leonard Lindelöf, Lead paragraph [in-branch 2]
Quote, Includes: Ernst Leonard Lindelöf, Lead paragraph [in-branch 2]
complex analysis
View the Source
George Pólya (Wikipedia)
Includes: George Pólya, Lead paragraph [in-branch 3]
Quote, Includes: George Pólya, Lead paragraph [in-branch 3]
complex analysis
View the Source
Eugenio Elia Levi (Wikipedia)
Includes: Eugenio Elia Levi, Lead paragraph [in-branch 2]
Quote, Includes: Eugenio Elia Levi, Lead paragraph [in-branch 2]
functions of several complex variables
View the Source
Wikipedia: Gösta Mittag-Leffler
Includes: Gösta Mittag-Leffler, Lead
Quote, Includes: Gösta Mittag-Leffler, Lead
His mathematical contributions are connected chiefly with the theory of functions that today is called complex analysis.
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Wikipedia: Constantin Carathéodory
Includes: Constantin Carathéodory, Lead
Quote, Includes: Constantin Carathéodory, Lead
He made significant contributions to real and complex analysis, the calculus of variations, and measure theory.
View the Source
Abel's test, Wikipedia
Includes: Abel's Test, Lead paragraph
Quote, Includes: Abel's Test, Lead paragraph
There are two slightly different versions of Abel's test, one is used with series of real numbers, and the other is used with power series in complex analysis.
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Wolfram MathWorld: Complex Plane
Wolfram MathWorldIncludes: Complex Plane, Subject classifications
Quote, Includes: Complex Plane, Subject classifications
Calculus and Analysis Complex Analysis
View the Source
Wolfram MathWorld: Barnes G-Function
Wolfram MathWorldIncludes: Barnes G-Function, Subject classifications
Quote, Includes: Barnes G-Function, Subject classifications
Calculus and Analysis Complex Analysis
View the Source
Wolfram MathWorld: Schwarz Reflection Principle
Wolfram MathWorldIncludes: Schwarz Reflection Principle, Subject classifications
Quote, Includes: Schwarz Reflection Principle, Subject classifications
Calculus and Analysis Complex Analysis
View the Source
Weierstrass preparation theorem, Wikipedia
Includes: Weierstrass Preparation Theorem, Lead section
Quote, Includes: Weierstrass Preparation Theorem, Lead section
Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point
View the Source
Möbius Transformation (Wikipedia)
Includes: Möbius Transformation, Lead section
Quote, Includes: Möbius Transformation, Lead section
In geometry and complex analysis, a Möbius transformation of the complex plane
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Joukowsky transform (Wikipedia)
Includes: Joukowsky transform, Lead section
Quote, Includes: Joukowsky transform, Lead section
transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles
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MacTutor biography of Felix Klein, St Andrews
Includes: Felix Klein, Biography
Quote, Includes: Felix Klein, Biography
results in function theory
View the Source
MacTutor biography of Édouard Goursat, St Andrews
Includes: Édouard Goursat, Summary
Quote, Includes: Édouard Goursat, Summary
Édouard Goursat was a French mathematician who is best known for his version of the Cauchy-Goursat theorem stating that the integral of a function round a simple closed contour is zero if the function is analytic inside the contour.
View the Source
Wolfram MathWorld: Riemann's Removable Singularity Theorem -- from Wolfram MathWorld
Wolfram MathWorldIncludes: Riemann's Removable Singularity Theorem, Subject classifications
Quote, Includes: Riemann's Removable Singularity Theorem, Subject classifications
Calculus and Analysis Complex Analysis
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Edmund Landau, Biography (MacTutor History of Mathematics)
Includes: Edmund Landau, Summary
Quote, Includes: Edmund Landau, Summary
Landau gave the first systematic presentation of analytic number theory and wrote important works on the theory of analytic functions of a single variable.
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Hermann Hankel, Biography (MacTutor History of Mathematics)
Includes: Hermann Hankel, Summary
Quote, Includes: Hermann Hankel, Summary
Hermann Hankel was a German mathematician who worked on the theory of complex numbers, the theory of functions and the history of mathematics. He is remembered for the Hankel transform and the Hankel matrix.
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Paul Koebe, Biography (MacTutor History of Mathematics)
Includes: Paul Koebe, Summary
Quote, Includes: Paul Koebe, Summary
Paul Koebe was a German mathematician who worked on complex functions. His colleagues found him brilliant but difficult.
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Walther von Dyck, Biography (MacTutor History of Mathematics)
Includes: Walther von Dyck, Summary
Quote, Includes: Walther von Dyck, Summary
Von Dyck made important contributions to function theory, group theory ( where a fundamental result on group presentations is named after him ) topology and potential theory.
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