Editing Felix Klein (section)

===Complex analysis===
Klein saw his work on [[complex analysis]] as his major contribution to mathematics, specifically his work on:
*The link between certain ideas of [[Bernhard Riemann|Riemann]] and [[invariant theory]],
*[[Number theory]] and [[abstract algebra]];
*[[Group theory]];
*[[Geometry]] in more than 3 dimensions and [[differential equations]], especially equations he invented, satisfied by [[elliptic modular function]]s and [[automorphic function]]s.

Klein showed that the [[modular group]] moves the fundamental region of the [[complex plane]] so as to [[tessellation|tessellate]] the plane. In 1879, he examined the action of [[PSL(2,7)]], considered as an image of the [[modular group]], and obtained an explicit representation of a [[Riemann surface]] now termed the [[Klein quartic]]. He showed that it  was a complex curve in [[projective space]], that its equation was ''x''{{sup|3}}''y''&nbsp;+&nbsp;''y''{{sup|3}}''z''&nbsp;+&nbsp;''z''{{sup|3}}''x''&nbsp;=&nbsp;0, and that its group of [[symmetry group|symmetries]] was [[PSL(2,7)]] of [[order (group theory)|order]] 168. His ''Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale'' (1882) treats complex analysis in a geometric way, connecting [[potential theory]] and [[conformal mapping]]s. This work drew on notions from [[fluid dynamics]].

Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on methods of [[Charles Hermite]] and [[Leopold Kronecker]], he produced similar results to those of Brioschi and later completely solved the problem by means of the [[icosahedral group]]. This work enabled him to write a series of papers on [[elliptic modular function]]s.

In his 1884 book on the [[icosahedron]], Klein established a theory of [[automorphic function]]s, associating algebra and geometry. [[Henri Poincaré|Poincaré]] had published an outline of his theory of automorphic functions in 1881, which resulted in a friendly rivalry between the two men. Both sought to state and prove a grand [[uniformization theorem]] that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it. He came up with his proof during an [[asthma attack]] at 2:30 A.M. on 23 March 1882.<ref>{{Cite journal |last=Abikoff |first=William |date=1981 |title=The Uniformization Theorem |url=https://www.jstor.org/stable/2320507 |journal=The American Mathematical Monthly |volume=88 |issue=8 |pages=574–592 |doi=10.2307/2320507 |jstor=2320507 |issn=0002-9890}}</ref>

Klein summarized his work on [[automorphic function|automorphic]] and [[elliptic modular function]]s in a four volume treatise, written with [[Robert Fricke]] over a period of about 20 years.