philosophy. Poincaré on discovery and his work on automorphic functions One of 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 and elliptic functions. Poincaré explains how he discovered Fuchsian functions: See also Automorphic factor Factor of automorphy Maass cusp form Automorphic Forms on GL(2), a book by H. Jacquet and Robert Langlands Jacobi form Notes References Henryk Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies in Mathematics), American Mathematical Society, Providence, RI Daniel Bump, "Automorphic Forms and Representations", 1998, Cambridge University Press Stephen Gelbart (1975), "Automorphic forms on Adele groups", External links Lie groups