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 QuestionWhich 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 DebateBernhard 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 AppliedBoth / Interdisciplinary 1 Connections
Associated With
Includes
Source Karl Weierstrass, Biography (MacTutor History of Mathematics)
Source Lars Ahlfors (Wikipedia)
Source Mergelyan's theorem (Wikipedia)
Source MacTutor History of Mathematics: Mikhail Alekseevich Lavrentev
Source Wikipedia: Mittag-Leffler's theorem
Source Möbius Transformation (Wikipedia)
Source Modular Lambda Function (Wikipedia)
Source Montel's theorem (Wikipedia)
Source Morera's theorem, Wikipedia
Source Mstislav Keldysh (Wikipedia)
Source Norman Levinson (Wikipedia)
Source Paul Julius Oswald Teichmüller (MacTutor)
Source Paul Koebe, Biography (MacTutor History of Mathematics)
Source Paul Montel (Wikipedia)
Source Phragmen-Lindelof principle (Wikipedia)
Source Resolvent Formalism (Wikipedia)
Source Riemann Sphere (Wikipedia)
Source Wikipedia: Riemann surface
Source Wolfram MathWorld: Riemann's Removable Singularity Theorem -- from Wolfram MathWorld
Source Rolf Nevanlinna (Wikipedia)
Source Runge's Theorem (Wikipedia)
Source Wolfram MathWorld: Schwarz Reflection Principle
Source Siméon Denis Poisson (Wikipedia)
Source Walter Rudin (Wikipedia)
Source Walther von Dyck, Biography (MacTutor History of Mathematics)
Source Weierstrass preparation theorem, Wikipedia
Source Wilhelm Wirtinger (Wikipedia)
Showing 51 to 78 of 78 in Includes, page 2 of 2.
Sources
1. Complex Analysis (Wikipedia)
Wikipedialead 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 sectionQuote, 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 summaryQuote, 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 sectionQuote, 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)
Theta function (Wikipedia)
Includes: Jacobi Theta Function, LeadQuote, 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, LeadQuote, 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 topicsQuote, 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 functionsQuote, 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, SummaryQuote, 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 workQuote, 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, ResearchQuote, 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.
View the Source Walter Rudin (Wikipedia)
Includes: Walter Rudin, LeadQuote, Includes: Walter Rudin, Lead
In addition to his contributions to complex and harmonic analysis
View the Source Giacinto Morera (Wikipedia)
Includes: Giacinto Morera, LeadQuote, 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, CareerQuote, 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, fieldQuote, 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 workQuote, Includes: Gábor Szegő, Biography, mathematical work
worked mainly in function theory
View the Source Harry Rauch (Wikipedia)
Includes: Harry Rauch, LeadQuote, Includes: Harry Rauch, Lead
who worked on complex analysis and differential geometry
View the Source Mstislav Keldysh (Wikipedia)
Includes: Mstislav Keldysh, Career, dissertationQuote, 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 variablesQuote, 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, LeadQuote, 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 theoryQuote, 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, dissertationQuote, 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 variableQuote, 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), SummaryQuote, 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 workQuote, 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 appointmentQuote, 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 chairQuote, 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, LeadQuote, Includes: Rolf Nevanlinna, Lead
made significant contributions to complex analysis
View the Source Wilhelm Wirtinger (Wikipedia)
Includes: Wilhelm Wirtinger, Lead, complex analysisQuote, 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, LeadQuote, Includes: Paul Montel, Lead
He researched mostly on holomorphic functions in complex analysis.
View the Source Allen Shields (Wikipedia)
Includes: Allen Shields, Lead, complex analysisQuote, 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, contributionsQuote, Includes: Oswald Teichmüller, Biography, contributions
He introduced quasi-conformal mappings and differential geometric methods into complex analysis.
View the Source Caspar Wessel (Wikipedia)
Includes: Caspar Wessel, LeadQuote, Includes: Caspar Wessel, Lead
geometrical interpretation of complex numbers
View the Source Montel's theorem (Wikipedia)
Includes: Montel's Theorem, Lead sentenceQuote, 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 sentenceQuote, 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 sentenceQuote, 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)
Borel-Caratheodory theorem - Wikipedia
Brennan Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Brennan Conjecture, Lead sentenceQuote, 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 sentenceQuote, Includes: Goodman's Conjecture, Lead sentence
ficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.
View the Source Wikipedia: Riemann surface
Includes: Riemann Surface, Lead sentenceQuote, 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 sentenceQuote, 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
View the Source Wikipedia: Mittag-Leffler's theorem
WikipediaIncludes: Mittag-Leffler Theorem, Lead sentenceQuote, Includes: Mittag-Leffler Theorem, Lead sentence
In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles.
View the Source Morera's theorem, Wikipedia
Includes: Morera's Theorem, Lead sentenceQuote, 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
View the Source Phragmen-Lindelof principle (Wikipedia)
Denjoy-Wolff theorem (Wikipedia)
Mergelyan's theorem (Wikipedia)
Includes: Mergelyan's Theorem, Lead sentenceQuote, 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.
View the Source Gauss-Lucas theorem (Wikipedia)
Bloch's theorem (complex analysis) (Wikipedia)
Includes: Bloch's Theorem (Complex Analysis), Lead sentenceQuote, 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, LeadQuote, Includes: Gösta Mittag-Leffler, Lead
His mathematical contributions are connected chiefly with the theory of functions that today is called complex analysis.
View the Source Wikipedia: Constantin Carathéodory
Includes: Constantin Carathéodory, LeadQuote, 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 paragraphQuote, 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.
View the Source Wolfram MathWorld: Complex Plane
Wolfram MathWorldIncludes: Complex Plane, Subject classificationsQuote, Includes: Complex Plane, Subject classifications
Calculus and Analysis Complex Analysis
View the Source Wolfram MathWorld: Barnes G-Function
Wolfram MathWorldIncludes: Barnes G-Function, Subject classificationsQuote, 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 classificationsQuote, Includes: Schwarz Reflection Principle, Subject classifications
Calculus and Analysis Complex Analysis
View the Source Weierstrass preparation theorem, Wikipedia
Includes: Weierstrass Preparation Theorem, Lead sectionQuote, 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 sectionQuote, Includes: Möbius Transformation, Lead section
In geometry and complex analysis, a Möbius transformation of the complex plane
View the Source Joukowsky transform (Wikipedia)
Includes: Joukowsky transform, Lead sectionQuote, Includes: Joukowsky transform, Lead section
transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles
View the Source MacTutor biography of Felix Klein, St Andrews
Includes: Felix Klein, BiographyQuote, Includes: Felix Klein, Biography
results in function theory
View the Source MacTutor biography of Édouard Goursat, St Andrews
Includes: Édouard Goursat, SummaryQuote, 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 classificationsQuote, Includes: Riemann's Removable Singularity Theorem, Subject classifications
Calculus and Analysis Complex Analysis
View the Source Edmund Landau, Biography (MacTutor History of Mathematics)
Includes: Edmund Landau, SummaryQuote, 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.
View the Source Hermann Hankel, Biography (MacTutor History of Mathematics)
Includes: Hermann Hankel, SummaryQuote, 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.
View the Source Paul Koebe, Biography (MacTutor History of Mathematics)
Includes: Paul Koebe, SummaryQuote, Includes: Paul Koebe, Summary
Paul Koebe was a German mathematician who worked on complex functions. His colleagues found him brilliant but difficult.
View the Source Walther von Dyck, Biography (MacTutor History of Mathematics)
Includes: Walther von Dyck, SummaryQuote, 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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