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 Complex Analysis
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Source Complex Analysis (Wikipedia)
Source Complex Analysis (Wikipedia)
Source Bloch's theorem (complex analysis) (Wikipedia)
Source Borel-Caratheodory theorem - Wikipedia
Source Brennan Conjecture (Wikipedia)
Source Casorati-Weierstrass theorem (Wikipedia)
Source Complex Number (Wikipedia)
Source Denjoy-Wolff theorem (Wikipedia)
Source Ernst Leonard Lindelöf (Wikipedia)
Source Eugenio Elia Levi (Wikipedia)
Source Gauss-Lucas theorem (Wikipedia)
Source George Pólya (Wikipedia)
Source Goodman's Conjecture (Wikipedia)
Source Hadamard three-lines theorem, Wikipedia
Source Hermann Schwarz (Wikipedia)
Source Joseph Liouville (Wikipedia)
Source Karl Weierstrass, Biography (MacTutor History of Mathematics)
Source Mergelyan's theorem (Wikipedia)
Source Wikipedia: Mittag-Leffler's theorem
Source Montel's theorem (Wikipedia)
Source Morera's theorem, Wikipedia
Source Phragmen-Lindelof principle (Wikipedia)
Source Resolvent Formalism (Wikipedia)
Source Riemann Sphere (Wikipedia)
Source Wikipedia: Riemann surface
Source Runge's Theorem (Wikipedia)
Source Siméon Denis Poisson (Wikipedia)
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.
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.
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
Riemann Sphere (Wikipedia)
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
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
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
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
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.
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.
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
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.
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
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.
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
Hermann Schwarz (Wikipedia)
Includes: Hermann Schwarz, Lead paragraph [in-branch]Quote, Includes: Hermann Schwarz, Lead paragraph [in-branch]
complex analysis
Joseph Liouville (Wikipedia)
Includes: Joseph Liouville, Lead paragraph [in-branch]Quote, Includes: Joseph Liouville, Lead paragraph [in-branch]
complex analysis
Siméon Denis Poisson (Wikipedia)
Includes: Simeon Denis Poisson, Lead paragraph [in-branch 2]Quote, Includes: Simeon Denis Poisson, Lead paragraph [in-branch 2]
complex analysis
Ernst Leonard Lindelöf (Wikipedia)
Includes: Ernst Leonard Lindelof, Lead paragraph [in-branch 2]Quote, Includes: Ernst Leonard Lindelof, Lead paragraph [in-branch 2]
complex analysis
George Pólya (Wikipedia)
Includes: George Polya, Lead paragraph [in-branch 3]Quote, Includes: George Polya, Lead paragraph [in-branch 3]
complex analysis
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
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