Special Functions

Mathematics, at its heart, is the identification and study of patterns and structures. In the field of Analysis, there is a strong focus on the study of functions and their properties. Several special functions have been observed to be prevalent in various settings, both in and out of maths, since the seventieth century (and in some cases prior to that). The goal of our project would be to utilise tools from real and complex analysis to explore several fundamental special functions which include:

• The Gamma function. Motivated by a desire to extend the notion of factorial to real numbers, the Gamma function was discovered by Euler in the 18^th century. It is difficult to avoid the Gamma function as it appears in mathematical physics, in areas and volumes of simple geometric shapes, and even in the field of number theory, amongst other places.
• The Beta function. Intimately connected to the Gamma function, and strongly related to the evaluation of relatively "polynomial like integrals", the Beta function was also studied extensively by Euler (amongst others). Much like the Gamma function, the Beta functions also appear in a myriad of topics which include probability theory and statistics.
• Hypergeometric series and functions. Hypergeometric series are relatively natural series which give rise to a special type of functions known as Hypergeometric functions. Astonishingly, almost all of the elementary functions in maths, including \(\sin(x), \cos(x), \exp{x}, \log(1+x) \) amongst other, are either hypergeometric functions or a ratio of them. Many other non-elementary we encounter in mathematical physics also have a representation as hypergeometric series.

Prerequisites (essential) : Analysis I, Linear Algebra I and Complex Analysis II.

• Andrews G. E, Askey R, and Roy R. Special Functions.
• Beals R, and Wong R. Special Functions: A Graduate Text.