Editing Automorphic form (section)

==Poincaré on discovery and his work on automorphic functions==
One of [[Henri Poincaré|Poincaré]]'s first discoveries in mathematics, dating to the 1880s, was automorphic forms. He named them Fuchsian functions, after the mathematician [[Lazarus Fuchs]], because Fuchs was known for being a good teacher and had researched on differential equations and the theory of functions. Poincaré actually developed the concept of these functions as part of his doctoral thesis. Under Poincaré's definition, an automorphic function is one which is analytic in its domain and is invariant under a discrete [[infinite group]] of linear fractional transformations. Automorphic functions then generalize both [[Trigonometric functions|trigonometric]] and [[elliptic function]]s.

Poincaré explains how he discovered Fuchsian functions:
{{blockquote|For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the [[hypergeometric series]]; I had only to write out the results, which took but a few hours.}}