Special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special.
The may be denoted , , , or depending on the context.
natural logarithm
The function may be denoted , , or ( is used in several European languages).
tangent
may be denoted , , , or .
Arctangent
History of special functions[edit]
Classical theory[edit]
While trigonometry and exponential functions were systematized and unified by the eighteenth century, the search for a complete and unified theory of special functions has continued since the nineteenth century. The high point of special function theory in 1800–1900 was the theory of elliptic functions; treatises that were essentially complete, such as that of Tannery and Molk,[3] expounded all the basic identities of the theory using techniques from analytic function theory (based on complex analysis). The end of the century also saw a very detailed discussion of spherical harmonics.
Changing and fixed motivations[edit]
While pure mathematicians sought a broad theory deriving as many as possible of the known special functions from a single principle, for a long time the special functions were the province of applied mathematics. Applications to the physical sciences and engineering determined the relative importance of functions. Before electronic computation, the importance of a special function was affirmed by the laborious computation of extended tables of values for ready look-up, as for the familiar logarithm tables. (Babbage's difference engine was an attempt to compute such tables.) For this purpose, the main techniques are:
Special functions in number theory[edit]
In number theory, certain special functions have traditionally been studied, such as particular Dirichlet series and modular forms. Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out of the monstrous moonshine theory.
Special functions of matrix arguments[edit]
Analogues of several special functions have been defined on the space of positive definite matrices, among them the power function which goes back to Atle Selberg,[6] the multivariate gamma function,[7] and types of Bessel functions.[8]
The NIST Digital Library of Mathematical Functions has a section covering several special functions of matrix arguments.[9]