Branches of Mathematics
Complex Analysis
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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.
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 Cross-Tradition Connections
Sources
1. Complex Analysis (Wikipedia)
Wikipedialead paragraphQuote, 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.
1. Complex Analysis (Wikipedia)
WikipediaHolomorphic functions sectionQuote, Holomorphic functions section
Complex analysis primarily studies holomorphic functions, which are differentiable functions of a complex variable.
1. Complex Analysis (Wikipedia)
WikipediaIncludes: Bernhard Riemann, History sectionQuote, Includes: Bernhard Riemann, History section
Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century.
1. Complex Analysis (Wikipedia)
WikipediaIncludes: Augustin-Louis Cauchy, History sectionQuote, 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.
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
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