List of mathematical functions
In
mathematics, several
functions or groups of functions are important enough to deserve their own names. This is a listing of pointers to those articles which explain these functions in more detail. There is a large theory of
special functions which developed out of
statistics and
mathematical physics. A modern, abstract point of view contrasts large
function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as
symmetry, or relationship to
harmonic analysis and
group representations. See also
orthogonal polynomial.
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.
*
Polynomials: Can be generated by addition and multiplication alone.
**
Linear function: First degree polynomial, graph is a straight line.
**
Quadratic function: Second degree polynomial, graph is a
parabola.
**
Cubic function: Third degree polynomial.
**
Quartic function: Fourth degree polynomial.
**
Quintic function: Fifth degree polynomial.
*
Rational functions: A ratio of two polynomials.
*
Power functions (with a rational power): A function of the form
xm/n.
**
Square root: Yields a number whose square is the given one (
x1/2).
Elementary transcendental functions
Transcendental functions are functions that are not algebraic.
*
Exponential function: raises a fixed number to a variable power.
*
Hyperbolic functions: formally similar to the trigonometric functions.
*
Logarithm: the inverses of exponential functions; useful to solve equations involving exponentials.
*
Power function: raises a variable number to a fixed power; also known as
Allometric function; note: if the power is a rational number it is not strictly a transcendental function.
*
Periodic functions
**
Trigonometric functions: sine, cosine, tangent, etc.; used in
geometry and to describe periodic phenomena. See also
Gudermannian function.
**
Sawtooth wave**
Square wave**
Triangle waveBasic special functions
*
Indicator function: maps
x to either 1 or 0, depending on whether or not
x belongs to some subset.
*
Step function: A finite
linear combination of
indicator functions of
half-open intervals.
**
Floor function: Largest integer less than or equal to a given number.
**
Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the
Dirac delta function.
**
Signum function: Returns only the sign of a number, as +1 or −1.
*
Absolute value: distance to the origin (zero point)
Number theoretic functions
*
Sigma function:
Sums of
powers of
divisors of a given
natural number.
*
Euler's totient function: Number of numbers
coprime to (and not bigger than) a given one.
*
Prime-counting function: Number of
primes 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.
Antiderivatives of elementary functions
*
Logarithmic integral function: Integral of the reciprocal of the logarithm, important in the
prime number theorem.
*
Exponential integral*
Error function: An integral important for
normal random variables.
**
Fresnel integral: related to the error function; used in
optics.
**
Dawson function: occurs in
probability.
Gamma and related functions
*
Gamma function: A generalization of the
factorial function.
*
Barnes G-function*
Beta function: Corresponding
binomial coefficient analogue.
*
Digamma function,
Polygamma function*
Incomplete beta function*
Incomplete gamma function*
K-function*
Multivariate gamma function: A generalization of the Gamma function useful in
multivariate statistics.
*
Student's t-distributionElliptic and related functions
*
Elliptic integrals: Arising from the path length of
ellipses; important in many applications. Related functions are the
quarter period and the
nome. Alternate notations include:
**
Carlson symmetric form**
Legendre form*
Elliptic functions: The inverses of elliptic integrals; used to model double-periodic phenomena. Particular types are
Weierstrass's elliptic functions and
Jacobi's elliptic functions.
*
Theta function* Closely related are the
modular forms, which include
**
J-invariant**
Dedekind eta functionBessel and related functions
*
Airy function*
Bessel functions: Defined by a
differential equation; useful in
astronomy,
electromagnetism, and
mechanics.
*
Bessel-Clifford function*
Legendre function: From the theory of
spherical harmonics.
*
Scorer's function*
Sinc function*
Hermite polynomials*
Chebyshev polynomialsRiemann zeta and related functions
*
Riemann zeta function: A special case of
Dirichlet series.
*
Dirichlet eta function: An allied function.
*
Hurwitz zeta function*
Legendre chi function*
Lerch Transcendent*
Polylogarithm and related functions:
**
Incomplete polylogarithm**
Clausen function**
Complete Fermi-Dirac integral, an alternate form of the polylogarithm.
**
Incomplete Fermi-Dirac integral**
Kummer's function*
Riesz functionHypergeometric and related functions
*
Hypergeometric functions: Versatile family of
power series.
*
Confluent hypergeometric function*
Associated Legendre polynomialsOther standard special functions
*
Dawson function*
Lambda function*
Lambert's W function: Inverse of
f(
w) =
w exp(
w).
*
Lame function*
Mittag-Leffler function*
Parabolic cylinder function*
Synchrotron functionMiscellaneous functions
*
Ackermann function: in the
theory of computation, a
computable function that is not
primitive recursive.
*
Dirac delta function: 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: is an
indicator function that matches 1 to rational numbers and 0 to irrationals. It is
nowhere continuous.
*
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: is an example of
continuous function that is nowhere
differentiableFunctions can be classified according to the properties they have. These properties describe the functions behaviour under certain conditions.
These properties concern the
domain, the
codomain and the
range of functions.
*
Bijective function: is both an
injective and a
surjection, and thus
invertible.
*
Composite function: is formed by the composition of two functions
f and
g, by mapping
x to
f(
g(
x)).
*
Constant function: has a fixed value regardless of arguments.
*
Empty function: whose domain equals the
empty set.
*
Inverse function: is declared by "doing the reverse" of a given function (e.g.
arcsine is the inverse of
sine).
*
Injective function: has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function.
*
Surjective function: has a
preimage for every element of the
codomain, i.e. the codomain equals the range. Also called a surjection or
onto function.
*
Identity function: maps any given element to itself.
*
Piecewise function: is defined by different expressions at different intervals.
Relative to an operator (c.q. a group)
These properties concern how the function is affected by
arithmetic operations on its operand.
*
Additive function: preserves the addition operation:
f(
x+
y) =
f(
x)+
f(
y).
*
Even function: is symmetric with respect to the
Y-axis. Formally, for each
x:
f(
x) =
f(−
x).
*
Odd function: is symmetric with respect to the
origin. Formally, for each
x:
f(−
x) = −
f(
x).
*
Subadditive function: for which the value of
f(
x+
y) is less than or equal to
f(
x)+
f(
y).
*
Superadditive function: for which the value of
f(
x+
y) is greater than or equal to
f(
x)+
f(
y).
Relative to a topology
*
Continuous function: in which
preimages of
open sets are open.
*
Nowhere continuous function: is not continuous at any point of its domain (e.g.
Dirichlet function).
*
Homeomorphism: is an
injective function that is also
continuous, whose
inverse is continuous.
Relative to an ordering
*
Monotonic function: does not reverse ordering of any pair.
* Strict
Monotonic function: preserves the given order.
Relative to the real/complex numbers
*
Analytic function: Can be defined locally by a
convergent power series.
*
Arithmetic function: A function from the positive
integers into the
complex numbers.
*
Differentiable function: Has a
derivative.
*
Holomorphic function:
Complex valued function of a complex variable which is differentiable at every point in its domain.
*
Entire function: A
holomorphic function whose domain is the entire
complex plane.
*
Special functions at EqWorld: The World of Mathematical Equations.
