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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

Analysis, Branches of Mathematics

Includes

Argument Principle, Theorems
Augustin-Louis Cauchy, Mathematicians
Source Complex Analysis (Wikipedia)
Bernhard Riemann, Mathematicians
Source Complex Analysis (Wikipedia)
Complex Number, Concepts
Source Complex Number (Wikipedia)
Goursat's Theorem, Theorems
Identity Theorem (Complex Analysis), Theorems
Karl Weierstrass, Mathematicians
Source Karl Weierstrass, Biography (MacTutor History of Mathematics)
Open Mapping Theorem (Complex Analysis), Theorems
Riemann Sphere, Mathematical Objects
Source Riemann Sphere (Wikipedia)
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
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