List of mathematical functions Totally Explained

Everything about List Of Functions totally explained

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 List of types of functions

Elementary functions

Elementary functions are functions built from basic operations (for example 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 $x^ !$ .

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.

Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.

Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it isn't 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 wave

Special functions

Basic 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-distribution

Elliptic 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:

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

Bessel 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 polynomials

Riemann 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
Spence's function

Riesz function

Hypergeometric and related functions

Hypergeometric functions: Versatile family of power series.

Confluent hypergeometric function

Associated Legendre polynomials

Meijer G-Function

Iterated exponential and related functions

Hyper operators

Iterated logarithm

Super-logarithms

Tetration

Lambert W function: Inverse of f(w) = w exp(w).

Other standard special functions

Lambda function

Lamé function

Mittag-Leffler function

Painleve transcendents

Parabolic cylinder function

Synchrotron function

Miscellaneous functions

Ackermann function: in the theory of computation, a computable function that isn't 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're equal, and 0 otherwise.

Minkowski's question mark function: Derivatives vanish on the rationals.

Weierstrass function: is an example of continuous function that's nowhere differentiableFurther Information

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