Special functions

Notations used for special functions

Functions with established international notations are the sine (${\displaystyle \sin }$), cosine (${\displaystyle \cos }$), exponential function (${\displaystyle \exp }$), and error function (${\displaystyle \operatorname {erf} }$ or ${\displaystyle \operatorname {erfc} }$).

Some special functions have several notations:

• The natural logarithm may be denoted ${\displaystyle \ln }$, ${\displaystyle \log }$, ${\displaystyle \log _{e}}$, or ${\displaystyle \operatorname {Log} }$ depending on the context.
• The tangent function may be denoted ${\displaystyle \tan }$, ${\displaystyle \operatorname {Tan} }$, or ${\displaystyle \operatorname {tg} }$ (used in several European languages).
• arctangent may be denoted ${\displaystyle \arctan }$, ${\displaystyle \operatorname {atan} }$, ${\displaystyle \operatorname {arctg} }$, or ${\displaystyle \tan ^{-1}}$.
• The Bessel functions may be denoted
  • ${\displaystyle J_{n}(x),}$
  • ${\displaystyle \operatorname {besselj} (n,x),}$
  • ${\displaystyle {\rm {BesselJ}}[n,x].}$

Subscripts are often used to indicate arguments, typically integers. In a few cases, the semicolon (;) or even backslash (\) is used as a separator for arguments. This may confuse the translation to algorithmic languages.

Superscripts may indicate not only a power (exponent), but some other modification of the function. Examples (particularly with trigonometric and hyperbolic functions) include:

• ${\displaystyle \cos ^{3}(x)}$ usually means ${\displaystyle (\cos(x))^{3}}$
• ${\displaystyle \cos ^{2}(x)}$ is typically ${\displaystyle (\cos(x))^{2}}$, but never ${\displaystyle \cos(\cos(x))}$
• ${\displaystyle \cos ^{-1}(x)}$ usually means ${\displaystyle \arccos(x)}$, not ${\displaystyle (\cos(x))^{-1}}$; this may cause confusion, since the meaning of this superscript is inconsistent with the others.

Evaluation of special functions

Most special functions are considered as a function of a complex variable. They are analytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to the Taylor series or asymptotic series are available. In addition, sometimes there exist relations with other special functions; a complicated special function can be expressed in terms of simpler functions. Various representations can be used for the evaluation; the simplest way to evaluate a function is to expand it into a Taylor series. However, such representation may converge slowly or not at all. In algorithmic languages, rational approximations are typically used, although they may behave badly in the case of complex argument(s).

Classical theory

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

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:

• numerical analysis, the discovery of infinite series or other analytical expressions allowing rapid calculation; and
• reduction of as many functions as possible to the given function.

More theoretical questions include: asymptotic analysis; analytic continuation and monodromy in the complex plane; and symmetry principles and other structural equations.

Twentieth century

The twentieth century saw several waves of interest in special function theory. The classic Whittaker and Watson (1902) textbook^[4] sought to unify the theory using complex analysis; the G. N. Watson tome A Treatise on the Theory of Bessel Functions pushed the techniques as far as possible for one important type, including asymptotic results.

The later Bateman Manuscript Project, under the editorship of Arthur Erdélyi, attempted to be encyclopedic, and came around the time when electronic computation was coming to the fore and tabulation ceased to be the main issue.

Contemporary theories

The modern theory of orthogonal polynomials is of a definite but limited scope. Hypergeometric series, observed by Felix Klein to be important in astronomy and mathematical physics,^[5] became an intricate theory, requiring later conceptual arrangement. Lie group representations give an immediate generalization of spherical functions; from 1950 onwards substantial parts of classical theory were recast in terms of Lie groups. Further, work on algebraic combinatorics also revived interest in older parts of the theory. Conjectures of Ian G. Macdonald helped open up large and active new fields with a special function flavour. Difference equations have begun to take their place beside differential equations as a source of special functions.

Bibliography

• Andrews, George E.; Askey, Richard; Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications. Vol. 71. Cambridge University Press. ISBN 978-0-521-62321-6. MR 1688958.
• Terras, Audrey (2016). Harmonic analysis on symmetric spaces – Higher rank spaces, positive definite matrix space and generalizations (second ed.). Springer Nature. ISBN 978-1-4939-3406-5. MR 3496932.
• Whittaker, E. T.; Watson, G. N. (1996-09-13). A Course of Modern Analysis. Cambridge University Press. ISBN 978-0-521-58807-2.
• N. N. Levedev (Translated & Edited by Richard A. Sliverman): Special Functions & Their Applications, DOVER, ISBN 978-0-486-60624-8 (1972). # Originally published from Prentice-Hall Inc.(1965).
• Nico M. Temme: Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley-Interscience,ISBN 9781118030813 (1996).
• Yury A. Brychkov: Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas, CRC Press, ISBN 978-1-58488-956-4 (2008).
• W. W. Bell: Special Functions : for Scientists and Engineers, Dover, ISBN 978-0-486-43521-3 (2004).
• Claude Brezinski and Michela Redivo-Zaglia: The Birth and Early Developments of Orthogonal Polynomials: A Chronological History, SIAM, ISBN 978-1-61197-850-6 (2025).

Numerical calculation method of function value

• Shanjie Zhang and Jian-Ming Jin: Computation of Special Functions, Wiley-Interscience, ISBN 978-0-471-11963-0 (1996).
• William J. Thompson: Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners; With Programs in C and Mathematica, Wiley-Interscience, ISBN 978-0-471-00260-4 (March, 1997).
• William J. Thompson: Atlas for Computing Mathematical Functions: An illustrated Guide for Practitioners; With Programs in Fortran 90 and Mathematica, Wiley-Interscience, ISBN 978-0-471-18171-2 (June, 1997).
• Amparo Gil, Javier Segura and Nico M. Temme: Numerical Methods for Special Functions, SIAM, ISBN 978-0-89871-634-4 (2007).
• Nico M. Temme: "Numerical aspects of special functions". Acta Numerica (2008), pp. 1–101