# List of mathematical functions

In mathematics, several functions are important enough to deserve their own name. 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 trigonometry, and then the needs of 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.

## Contents

## Classes of functions

Functions can classified according to the properties they have.

- Additive function: The value of a product equals the sum of the values of the factors.
- Analytic function: Can be defined locally by a convergent power series.
- Arithmetic function: A function from the positive integers into the complex numbers.
- Bijection: Is both an injective and a surjection.
- Composition function: For two functions
*f*and*g*, maps*x*to*f*(*g*(*x*)). - Continuous function: Preimages of open sets are open.
- Differentiable function: Has a derivative.
- Entire function: A holomorphic function whose domain is the entire complex plane.
- Even function:
*f*(*x*) =*f*(−*x*). Is symmetric with respect to the*Y*-axis. - Holomorphic function: Complex valued function of a complex variable which is complex differentiable at every point in its domain.
- Homeomorphism: One-to-one function continuous function, whose inverse is continuous.
- Injection, injective function: Distinct arguments have distinct values. Also called a one-to-one function.
- Inverse function: "Does the reverse" of a given function.
- Monotonic function: Order preserving (or reversing).
- Odd function:
*f*(*x*) = −*f*(*x*). Is symmetric with respect to the origin. - One-to-one function: Distinct arguments have distinct values. Also called an injection or injective function.
- Onto function: Every element of the codomain has a preimage. Also called a surjection or surjective function.
- Subadditive function: The value of a sum is less than or equal to the sum of the values of the summands.
- Superadditive function: The value of a sum is greater than or equal to the sum of the values of the summands.
- Surjection, surjective function: Every element of the codomain has a preimage. Also called an onto function.

## Elementary functions

- Absolute value: Leaves positive numbers alone, multiplies negative numbers by −1 to make them positive.
- Empty function: Domain equals the empty set.
- 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.
- Identity function: Maps a given element to itself.
- Indicator function: Maps
*x*to either 1 or 0, depending on whether, x is or is not in some set. - Signum function: Returns only the sign of a number, as +1 or −1.
- Step function: A finite linear combination of indicator functions of half-open intervals.

### Polynomials

Polynomials: can be generated by addition and multiplication alone.

- Constant function: Zero degree polynomial, fixed value regardless of arguments.
- 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.

### Elementary periodic functions

### Elementary transcendental functions

- 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.
- Square root: yields a number whose square is the given one.
- Trigonometric functions: sine, cosine, tangent, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.

## Special functions

### 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 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-distribution

- 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:
- 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

- Airy function
- Bessel functions: Defined by a differential equation; useful in astronomy, electromagnetism, and mechanics.
- Legendre function: From the theory of spherical harmonics.
- Scorer's function
- Sinc function

- 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 function

- Hypergeometric functions: Versatile family of power series.
- Confluent hypergeometric function
- Associated Legendre polynomials

### Other 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 function

## 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.

## Miscellaneous

- Ackermann function: in the theory of computation, a recursive 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: Nowhere continuous.
- Question mark function: Derivatives vanish on the rationals.
- Weierstrass function: Continuous, nowhere differentiable

## External links

- Special functions at EqWorld: The World of Mathematical Equations.

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