Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the mathematical analysis, functional analysis, physics and other applications. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Analysis has its beginnings in the rigorous formulation of Calculus. For functional analysis as used in psychology see the Functional analysis (psychology article Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.
There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. In Mathematics, several functions or groups of functions are important enough to deserve their own names In particular, elementary functions are also considered as special functions. This article discusses the concept of elementary functions in differential algebra
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Tables of special functions
Many special functions appear as solutions of differential equations or integrals of elementary functions. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space This article discusses the concept of elementary functions in differential algebra Therefore, tables of integrals [1] usually include description of special functions, and tables of special functions [2] include most important integrals; at least, the integral representation of special functions.
Languages for analytical calculus such as Mathematica[3] usually recognize the majority of special functions. Mathematica is a computer program used widely in scientific engineering and mathematical fields Not all such systems have efficient algorithms for the evaluation, especially in the complex plane.
Notations used in special functions
In the most of cases, the standard notation is used for indication of a special function: the name of function (printed with Roman font), subscripts, if any, open parenthesis, and then arguments, separated with comma. Such a notation allows easy translation of the expressions to algorithmic languages avoiding ambiguities. Functions with established international notations are sin, cos, exp, erf, erfc.
Sometimes, a special function has several names. The natural logarithm can be called as Log, log or ln, dependently on the context. The tangent may be called as Tan, tan or tg (especially in Russian literature); arctangent can be called atan, arctg, tan − 1. Function of Bessel can be called just 
; usually, 
, 
, ![~ {\rm BesselJ}[n,x]~](../../../../math/5/4/d/54dd1f04507b14ca7af143b62fdf3d19.png)
refer to the same function.
Often the subscriptors are used to indicate argument(s), which is(are) usually supposed to be integer. In few faces, the semicolon (;) or even backslash (\) is used as separator. Then, the translation to algorithmic languages allows ambiguity and may lead to confusions. Ambiguity (Am-big-u-i-ty is the property of being ambiguous, where a Word, term notation sign Symbol, Phrase, sentence, or any
Superscript may indicate not only exponential, but modification of function. For example, 
, 
may indicate 
, 
, 
(or 
), respectively; but 
almost never means 
.
Evaluation of special functions
Most of special functions are considered as a functions of complex variable(s). They are analytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to the Taylor or asymptotic series are available. This article is about both real and complex analytic functions In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which In addition, sometimes there exist relations with other special functions; a complicated special function can be expressed in terms of simple functions. Various representations can be used for the evaluation; the simplest way to evaluate a function is to expand it into a Taylor series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives However, such representation may converge slowly if at all. In algorithmic languages, usually, the rational approximations are used, although, the rational approximations may be not so good in the case of complex argument(s).
History of special functions
Classical theory
While trigonometry can be codified, as was clear already to expert mathematicians of the eighteenth century (if not before), the search for a complete and unified theory of special functions has continued since the nineteenth century. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar The high point of the 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. In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic They were based on complex analysis techniques. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex
From that time onwards it would be assumed that analytic function theory, which had already unified the trigonometric and exponential functions, was a fundamental tool. This article is about both real and complex analytic functions The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) The end of the century also saw a very detailed discussion of spherical harmonics. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of
Changing and fixed motivations
Of course the wish for a broad theory including as many as possible of the known special functions has its intellectual appeal, but it is worth noting other reasons for wanting it. For a long time the special functions were in the particular province of applied mathematics; applications to the physical sciences and engineering determined the relative importance of functions. Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains In the days before the electronic computer, the ultimate compliment to a special function was the computation, by hand, of extended tables of its values. A computer is a Machine that manipulates data according to a list of instructions. Before Calculators were cheap and plentiful people would use mathematical tables &mdashlists of numbers showing the results of calculation with varying arguments&mdash to simplify This was a capital-intensive process, intended to make the function available by look-up, as for the familiar logarithm tables. In Computer science, a lookup table is a Data structure, usually an Array or Associative array, often used to replace a runtime computation with The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base The aspects of the theory that then mattered might then be two:
- for numerical analysis, discovery of infinite series or other analytical expression allowing rapid calculation; and
- reduction of as many functions as possible to the given function. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, an analytical expression (or expression in analytical form) is a Mathematical expression, constructed using well-known operations that lend
In contrast, one might say, there are approaches typical of the interests of pure mathematics: asymptotic analysis, analytic continuation and monodromy in the complex plane, and the discovery of symmetry principles and other structure behind the façade of endless formulae in rows. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. In Mathematics, monodromy is the study of how objects from Mathematical analysis, Algebraic topology and algebraic and Differential geometry In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or There is not a real conflict between these approaches, in fact.
Twentieth century
The twentieth century saw several waves of interest in special function theory. The twentieth century of the Common Era began on The classic Whittaker and Watson 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. Whittaker and Watson is the informal name of a book formally entitled A Course of Modern Analysis, written by E Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex (George Neville Watson ( 31 January 1886 – 2 February 1965) was an English mathematician a noted master in the application of Complex
The later Bateman manuscript project, under the editorship of Arthur Erdélyi, attempted to be encyclopedic, and came at about the time when electronic computation was changing the motivations. The Bateman Manuscript Project was a major effort at collation and encyclopedic compilation of the mathematical theory of Special functions. Arthur Erdélyi ( October 2, 1908 – December 12, 1977) was a Hungarian-born British Mathematician. Tabulation was no longer the main issue.
Contemporary theories
The modern theory of orthogonal polynomials is of a definite but limited scope. In Mathematics, an orthogonal polynomial sequence is an infinite sequence of real Polynomials p_0\ p_1\ p_2\ \ldots Hypergeometric series became an intricate theory, in need of later conceptual arrangement. In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function Lie groups, and in particular their representation theory, explain what a spherical function can be in general; from 1950 onwards substantial parts of classical theory could be recast in Lie group terms. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Spherical function can refer to Spherical harmonics Zonal spherical function Further, the work on algebraic combinatorics also revived interest in older parts of the theory. Algebraic combinatorics is an area of Mathematics that employs methods of Abstract algebra, notably Group theory and Representation theory, in Conjectures of Ian G. Macdonald helped to open up large and active new fields with the typical special function flavour. Ian G Macdonald (born 1928 in London, England) is a British mathematician known for his contributions to Symmetric functions Special functions Difference equations have begun to take their place besides differential equations as a source for special functions. "Difference equation" redirects here It should not be confused with a Differential equation. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the
Special functions in number theory
In number theory certain special functions have traditionally been studied, such as particular Dirichlet series and modular forms. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and 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. In Mathematics, monstrous moonshine is a term devised by John Horton Conway and Simon P
See also
References
- ^ Gradshtein, Israel Solomonovich; Iosif Moiseevich Ryzhik. In Mathematics, several functions or groups of functions are important enough to deserve their own names This is a list of special function eponyms in Mathematics, to cover the theory of Special functions, the Differential equations they satisfy named . Table of integrals, sums, series and products. Academic press.
- ^ Abramovitz, Milton; Irene Stegun. Table of mathematical functions.
- ^ List of special functions in Mathematica
External links
- Special functions, Special functions calculator.
- Special functions at EqWorld: The World of Mathematical Equations.
- Abramowitz and Stegun, handbook on special functions. Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U
- http://www.amazon.ca/exec/obidos/ASIN/0122947576, Gradshtein, Ryshik
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