In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 out of
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
and
mathematical physics
Mathematical physics is 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 the de ...
. A modern, abstract point of view contrasts large
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
s, which are infinite-dimensional and within which most functions are "anonymous", with special functions picked out by properties such as
symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
, or relationship to
harmonic analysis and
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s.
See also
List of types of functions
Elementary functions
Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...)
Algebraic functions
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
*
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
**
Constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value.
Basic properties
As a real-valued function of a real-valued argument, a constant function has the general form or just For example, ...
: polynomial of degree zero, graph is a horizontal straight line
**
Linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
: First degree polynomial, graph is a straight line.
**
Quadratic function
In mathematics, a quadratic function of a single variable (mathematics), variable is a function (mathematics), function of the form
:f(x)=ax^2+bx+c,\quad a \ne 0,
where is its variable, and , , and are coefficients. The mathematical expression, e ...
: Second degree polynomial, graph is a
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
.
**
Cubic function: Third degree polynomial.
**
Quartic function
In algebra, a quartic function is a function (mathematics), function of the form
:f(x)=ax^4+bx^3+cx^2+dx+e,
where ''a'' is nonzero,
which is defined by a polynomial of Degree of a polynomial, degree four, called a quartic polynomial.
A ''qu ...
: Fourth degree polynomial.
**
Quintic function: Fifth degree polynomial.
*
Rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s: A ratio of two polynomials.
*
''n''th root
**
Square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
: Yields a number whose square is the given one.
**
Cube root
In mathematics, a cube root of a number is a number that has the given number as its third power; that is y^3=x. The number of cube roots of a number depends on the number system that is considered.
Every real number has exactly one real cub ...
: Yields a number whose cube is the given one.
Elementary transcendental functions
Transcendental functions are functions that are not algebraic.
*
Exponential function: raises a fixed number to a variable power.
*
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 ...
s: formally similar to the
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
.
**
Inverse hyperbolic functions
In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangen ...
:
inverses of the
hyperbolic functions, analogous to the
inverse circular functions.
*
Logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
s: the inverses of exponential functions; useful to solve equations involving exponentials.
**
Natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
**
Common logarithm
In mathematics, the common logarithm (aka "standard logarithm") is the logarithm with base 10. It is also known as the decadic logarithm, the decimal logarithm and the Briggsian logarithm. The name "Briggsian logarithm" is in honor of the British ...
**
Binary logarithm
In mathematics, the binary logarithm () is the exponentiation, power to which the number must be exponentiation, raised to obtain the value . That is, for any real number ,
:x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n.
For example, th ...
*
Power functions: raise a variable number to a fixed power; also known as
Allometric functions; note: if the power is a rational number it is not strictly a transcendental function.
*
Periodic function
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
s
**
Trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s:
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 opposite th ...
,
cosine
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 opposite that ...
,
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
,
cotangent
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
,
secant,
cosecant,
exsecant,
excosecant,
versine,
coversine,
vercosine,
covercosine,
haversine,
hacoversine,
havercosine,
hacovercosine,
Inverse trigonometric functions
In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted Domain of a functi ...
etc.; used in
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and to describe periodic phenomena. See also
Gudermannian function
In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwe ...
.
Special functions
Piecewise special functions
Arithmetic functions
*
Sigma function:
Sums of
powers of
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
s of a given
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
.
*
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
: Number of numbers
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
to (and not bigger than) a given one.
*
Prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ).
A symmetric variant seen sometimes is , which is equal ...
: Number of
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s less than or equal to a given number.
*
Partition function: Order-independent count of ways to write a given positive integer as a sum of positive integers.
*
Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n.
*
Prime omega functions
*
Chebyshev functions
*
Liouville function, λ(''n'') = (–1)
Ω(''n'')
*
Von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Mang ...
, Λ(''n'') = log ''p'' if ''n'' is a positive power of the prime ''p''
*
Carmichael function
Antiderivatives of elementary functions
*
Logarithmic integral function
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theory, number theoretic significance. In particular, according to the prime number the ...
: Integral of the reciprocal of the logarithm, important in the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by p ...
.
*
Exponential integral
In mathematics, the exponential integral Ei is a special function on the complex plane.
It is defined as one particular definite integral of the ratio between an exponential function and its argument.
Definitions
For real non-zero values of&nb ...
*
Trigonometric integral
In mathematics, trigonometric integrals are a indexed family, family of nonelementary integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
\operatorname(x) = \int_0^x\frac\,dt
\operato ...
: Including Sine Integral and Cosine Integral
*
Inverse tangent integral
*
Error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as:
\operatorname z = \frac\int_0^z e^\,\mathrm dt.
The integral here is a complex Contour integrat ...
: An integral important for
normal random variables.
**
Fresnel integral: related to the error function; used in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
.
**
Dawson function: occurs in
probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
.
**
Faddeeva function
Gamma and related functions
*
Gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
: A generalization of the
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
function.
*
Barnes G-function
*
Beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^ ...
: Corresponding
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
analogue.
*
Digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(z) = \frac\ln\Gamma(z) = \frac.
It is the first of the polygamma functions. This function is Monotonic function, strictly increasing a ...
,
Polygamma function
*
Incomplete beta function
*
Incomplete gamma function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals.
Their respective names stem from their integral definitions, whic ...
*
K-function
*
Multivariate gamma function: A generalization of the Gamma function useful in
multivariate statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable, i.e., '' multivariate random variables''.
Multivariate statistics concerns understanding the differ ...
.
*
Student's t-distribution
In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
*
Pi function
Elliptic and related functions
Bessel and related functions
Riemann zeta and related functions
{{columns-list, colwidth=20em,
*
Riemann zeta function: A special case of
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 anal ...
.
*
Riemann Xi function
*
Dirichlet eta function: An allied function.
*
Dirichlet beta function
*
Dirichlet L-function
In mathematics, a Dirichlet L-series is a function of the form
:L(s,\chi) = \sum_^\infty \frac.
where \chi is a Dirichlet character and s a complex variable with real part greater than 1 . It is a special case of a Dirichlet series. By anal ...
*
Hurwitz zeta function
*
Legendre chi function
*
Lerch transcendent
*
Polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
and related functions:
**
Incomplete polylogarithm
**
Clausen function
**
Complete Fermi–Dirac integral, an alternate form of the polylogarithm.
**
Dilogarithm
**
Incomplete Fermi–Dirac integral
**
Kummer's function
*
Riesz function
Hypergeometric and related functions
*
Hypergeometric function
In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
s: Versatile family of
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
.
*
Confluent hypergeometric function
*
Associated Legendre functions
*
Meijer G-function
*
Fox H-function
Iterated exponential and related functions
*
Hyper operators
*
Iterated logarithm
*
Pentation
*
Super-logarithms
*
Tetration
In mathematics, tetration (or hyper-4) is an operation (mathematics), operation based on iterated, or repeated, exponentiation. There is no standard mathematical notation, notation for tetration, though Knuth's up arrow notation \uparrow \upa ...
Other standard special functions
*
Lambert W function: Inverse of ''f''(''w'') = ''w'' exp(''w'').
*
Lamé function
*
Mathieu function
*
Mittag-Leffler function
*
Painlevé transcendents
*
Parabolic cylinder function
*
Arithmetic–geometric mean
Miscellaneous functions
*
Ackermann function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total function, total computable function that is not Primitive recursive function, primitive recursive. ...
: in the
theory of computation, a
computable function
Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precis ...
that is not
primitive recursive.
*
Dirac delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
: everywhere zero except for ''x'' = 0; total integral is 1. Not a function but a
distribution, but sometimes informally referred to as a function, particularly by physicists and engineers.
*
Dirichlet function
In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number).
\mathb ...
: is an
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
that matches 1 to
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...
and 0 to
irrationals. It is
nowhere continuous.
*
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
*
Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
*
Minkowski's question mark function: Derivatives vanish on the rationals.
*
Weierstrass function
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiab ...
: is an example of
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
that is nowhere
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
See also
*
List of types of functions
*
Test functions for optimization
In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as Rate of convergence, convergence rate, precision, robustness and general performance.
Here some test ...
*
List of mathematical abbreviations
*
List of special functions and eponyms
External links
Special functions: A programmable special functions calculator.
at EqWorld: The World of Mathematical Equations.
Functions
Functions
Functions
pl:Funkcje elementarne