SPECIAL FUNCTIONS

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. In particular, elementary functions are also considered as special functions.

The high point of special function theory in the period 1850-1900 was the theory of elliptic functions; treatises that were essentially complete, such as that of Tannery and Molk, could be written as handbooks to all the basic identities of the theory. They were based on techniques from complex analysis. The twentieth century saw several waves of interest in special function theory. The classic Whittaker and Watson (1902) textbook sought to unify the theory by using complex variables; the G. N. Watson tome A Treatise on the Theory of Bessel Functions pushed the techniques as far as possible for one important type that particularly admitted asymptotics to be studied.

L1: Bessel’s Equation - Recurrence Relations of Bessel’s Functions - Generating Function

L2: Equations Reducible to Bessel’s Equations

L3: Ber and Bei Functions

L4: Legendre’s Equation - Rodrigue’s Formula - Generating Function for Pn(x) - Recurrence Relations of Legendre’s Functions - Orthogonality of Legendre’s Polynomials - Expansion of a Function in a Series of Legendre’s Polynomials