Special functions are particular
mathematical function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
s that have more or less established names and notations due to their importance in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
,
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
,
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, or other applications.
The term is defined by consensus, and thus lacks a general formal definition, but the
List of mathematical functions
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed ...
contains functions that are commonly accepted as special.
Tables of special functions
Many special functions appear as solutions of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s or
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s of
elementary functions
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
. Therefore, tables of integrals
usually include descriptions of special functions, and tables of special functions
[
] include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s and
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s, as well as certain topics in
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
.
Symbolic computation engines usually recognize the majority of special functions.
Notations used for special functions
Functions with established international notations are the
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
(
),
cosine (
),
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
(
), and
error function (
or
).
Some special functions have several notations:
* The
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
may be denoted
,
,
, or
depending on the context.
* The
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
function may be denoted
,
, or
(
is used mainly in
Russian
Russian(s) refers to anything related to Russia, including:
*Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries
*Rossiyane (), Russian language term for all citizens and peo ...
and
Bulgarian
Bulgarian may refer to:
* Something of, from, or related to the country of Bulgaria
* Bulgarians, a South Slavic ethnic group
* Bulgarian language, a Slavic language
* Bulgarian alphabet
* A citizen of Bulgaria, see Demographics of Bulgaria
* Bul ...
literature).
*
Arctangent may be denoted
,
,
, or
.
* The
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
s may be denoted
**
**
**
Subscripts are often used to indicate arguments, typically integers. In a few cases, the semicolon (;) or even backslash (\) is used as a separator. In this case, the translation to algorithmic languages admits
ambiguity
Ambiguity is the type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty. It is thus an attribute of any idea or statement ...
and may lead to confusion.
Superscripts may indicate not only exponentiation, but modification of a function. Examples (particularly with
trigonometric
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
and
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s) include:
*
usually means
*
is typically
, but never
*
usually means
, not
; this one typically causes the most 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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
variable. They are
analytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
or
asymptotic series In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
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).
History of special functions
Classical theory
While
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
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. 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
Tanning may refer to:
*Tanning (leather), treating animal skins to produce leather
*Sun tanning, using the sun to darken pale skin
**Indoor tanning, the use of artificial light in place of the sun
**Sunless tanning, application of a stain or dye t ...
and
Molk, could be written as handbooks to all the basic identities of the theory. They were based on techniques from
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
.
From that time onward it would be assumed that
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
theory, which had already unified the trigonometric and
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
s, was a fundamental tool. The end of the century also saw a very detailed discussion of
spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
s.
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 motivations. For a long time, the special functions were in the particular province of
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
; applications to the physical sciences and engineering determined the relative importance of functions. In the days before the
electronic computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These program ...
, the ultimate compliment to a special function was the computation, by hand, of extended
tables of its values. This was a capital-intensive process, intended to make the function available by
look-up, as for the familiar
logarithm tables. The aspects of the theory that then mattered might then be two:
* for
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, discovery of
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
or other
analytical expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
allowing rapid calculation; and
* reduction of as many functions as possible to the given function.
In contrast, one might say, there are approaches typical of the interests of
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
:
asymptotic analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...
,
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
and
monodromy in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, and the discovery of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
principles and other structure behind the façade of endless formulae in rows. 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 classic ''
Whittaker and Watson'' (1902) textbook sought to unify the theory by using
complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
s; 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.
The later
Bateman Manuscript Project, under the editorship of
Arthur Erdélyi
Arthur Erdélyi FRS, FRSE (2 October 1908 – 12 December 1977) was a Hungarian-born British mathematician. Erdélyi was a leading expert on special functions, particularly orthogonal polynomials and hypergeometric functions.
Biography
He ...
, 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
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
is of a definite but limited scope.
Hypergeometric series, observed by
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
to be important in
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, became an intricate theory, in need of later conceptual arrangement.
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s, and in particular their
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, explain what a
spherical function can be in general; from 1950 onwards substantial parts of classical theory could be recast in terms of Lie groups. Further, work on
algebraic combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algeb ...
also revived interest in older parts of the theory. Conjectures of
Ian G. Macdonald helped to open up large and active new fields with the typical special function flavour.
Difference equations have begun to take their place besides
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s as a source for special functions.
Special functions in number theory
In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, certain special functions have traditionally been studied, such as particular
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analy ...
and
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s. Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out of the
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
theory.
Special functions of matrix arguments
Analogues of several special functions have been defined on the space of
positive definite matrices
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a c ...
, among them the power function which goes back to
Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarde ...
, the
multivariate gamma function, and types of
Bessel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
.
The
NIST
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
Digital Library of Mathematical Functions has a section covering several special functions of matrix arguments.
Researchers
*
George Andrews
*
Richard Askey
Richard Allen Askey (4 June 1933 – 9 October 2019) was an American mathematician, known for his expertise in the area of special functions. The Askey–Wilson polynomials (introduced by him in 1984 together with James A. Wilson) are on the t ...
*
Harold Exton
Harold Exton is a mathematician at University of Central Lancashire (called Preston Polytechnic while he was there) working on hypergeometric functions, who introduced the Hahn–Exton q-Bessel function
In mathematics, the Hahn–Exton ''q''-Bess ...
*
George Gasper
*
Wolfgang Hahn
Wolfgang Hahn (April 30, 1911 – January 10, 1998) was a German mathematician who worked on special functions, in particular orthogonal polynomials.
He introduced Hahn polynomials, Hahn difference, Hahn q-addition (or Jackson-Hahn-Cigler q-add ...
*
Mizan Rahman
*
Mourad E. H. Ismail
*
Tom Koornwinder
*
Waleed Al-Salam
*
Dennis Stanton
*
Theodore S. Chihara
*
James A. Wilson
*
Erik Koelink
*
Eric Rains
See also
*
List of mathematical functions
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed ...
*
List of special functions and eponyms
*
Elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
References
Bibliography
*
*
External links
*
National Institute of Standards and Technology
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
, United States Department of Commerce
''NIST Digital Library of Mathematical Functions''Archivedfrom the original on December 13, 2018.
*
Online calculator Online scientific calculator with over 100 functions (>=32 digits, many complex) (German language)
at ''EqWorld: The World of Mathematical Equations''
''Special functions and polynomials''by Gerard 't Hooft and Stefan Nobbenhuis (April 8, 2013)
Numerical Methods for Special Functions by A. Gil, J. Segura, N.M. Temme (2007).
* R. Jagannathan
(P,Q)-Special FunctionsSpecialfunctionswiki
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History of mathematics