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Hypergeometric functions, from Riemann till present. (English) Zbl 1405.33002
Ji, Lizhen (ed.) et al., Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds and Picard-Fuchs equations. Based on the conference, Institute Mittag-Leffler, Stockholm, Sweden, July 13–18, 2015. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-363-0/pbk). Advanced Lectures in Mathematics (ALM) 42, 1-19 (2018).
Summary: Hypergeometric functions form a family of classical functions that occur frequently within many areas of mathematics and its applications. They were first introduced by Euler, who already discovered a great many surprising properties. Gauss continued this study by considering hypergeometric functions as solutions of a second order differential equation in the complex plane, including their multivaluedness. Riemann took up Gauss’ study and made hypergeometric functions as prime example for his ideas on analytic continuation. It was also Riemann who named them Gauss hypergeometric functions. Although there exist many generalizations nowadays, we concentrate ourselves on these original functions. We briefly sketch Riemann’s ideas and give an overview of developments around Gauss’ hypergeometric function until recent times. There is an overlap of the first two sections with the author’s summer school in [Prog. Math. 260, 23–42 (2007; Zbl 1118.14012)].
For the entire collection see [Zbl 1398.14003].
##### MSC:
 33C05 Classical hypergeometric functions, $${}_2F_1$$ 01A55 History of mathematics in the 19th century 01A60 History of mathematics in the 20th century 30C35 General theory of conformal mappings