A way of associating unique objects to every point in a given Set. A function from to is an object
such that for every , there is a unique object . Examples of functions include , , ,
etc. The term Map is synonymous with function.

Poincaré remarked with regard to the proliferation of pathological functions, ``Formerly, when one
invented a new function, it was to further some practical purpose; today one invents them in order to make incorrect the
reasoning of our fathers, and nothing more will ever be accomplished by these inventions.''

*See also *Abelian Function, Absolute Value, Ackermann Function, Airy Functions, Algebraic
Function, Algebroidal Function, Alpha Function, Andrew's Sine, Anger Function, Apodization
Function, Apparatus Function, Argument (Function), Artin *L*-Function, Automorphic Function,
Bachelier Function, Barnes *G*-Function, Bartlett Function, Basset Function, Bateman
Function, Bei, Ber, Bernoulli Function, Bessel Function of the First Kind, Bessel Function of
the Second Kind, Bessel Function of the Third Kind, Beta Function, Beta Function (Exponential),
Binomial Coefficient, Blackman Function, Blancmange Function, Boolean Function, Bourget
Function, Boxcar Function, Brown Function, Cal, Cantor Function, Carmichael Function,
Carotid-Kundalini Function, Ceiling Function, Center Function, Central Beta Function,
Characteristic Function, Chebyshev Function, Circular Functions, Clausen Function, Comb
Function, Complete Functions, Complex Conjugate, Computable Function, Concave Function,
Confluent Hypergeometric Function, Confluent Hypergeometric Function of the First Kind, Confluent
Hypergeometric Function of the Second Kind, Confluent Hypergeometric Limit Function, Conical Function,
Connes Function, Constant Function, Contiguous Function, Continuous Function, Convex
Function, Copula, Cosecant, Cosine, Cosine Apodization Function, Cotangent, Coulomb
Wave Function, Coversine, Cube Root, Cubed, Cumulant-Generating Function, Cumulative
Distribution Function, Cunningham Function, Cylinder Function, Cylindrical Function, Debye
Functions, Decreasing Function, Dedekind Eta Function, Dedekind Function, Delta Function,
Digamma Function, Dilogarithm, Dirac Delta Function, Dirichlet Beta Function, Dirichlet Eta
Function, Dirichlet Function, Dirichlet Lambda Function, Distribution Function, Divisor Function,
Double Gamma Function, Doublet Function, En-Function,
Et-Function, Eigenfunction, Ein Function, Einstein Functions, Elementary
Function, Elliptic Alpha Function, Elliptic Delta Function, Elliptic Exponential Function, Elliptic
Function, Elliptic Functional, Elliptic Lambda Function, Elliptic Modular Function, Elliptic Theta
Function, Elsasser Function, Entire Function, Epstein Zeta Function, Erdös-Selfridge
Function, Erf, Error Function, Exponential Ramp, Euler *L*-Function,
Even Function, Exponential Function, Exponential Function (Truncated), Exponential Sum Function,
Exsecant, Floor Function, Fox's *H*-Function, Function Space,
*G*-Function, Gamma
Function, Gate Function, Gaussian Function, Gegenbauer Function, Generalized Function,
Generalized Hyperbolic Functions, Generalized Hypergeometric Function, Generating Function, Gordon
Function, Green's Function, Growth Function, Gudermannian Function, *H*-Function, Haar
Function, Hamming Function, Hankel Function, Hankel Function of the First Kind, Hankel Function of
the Second Kind, Hann Function, Hanning Function, Harmonic Function, Haversine, Heaviside
Step Function, Hecke *L*-Function, Hemicylindrical Function, Hemispherical Function, Heuman Lambda
Function, Hh Function, Hilbert Function, Holonomic Function, Homogeneous Function, Hurwitz
Zeta Function, Hyperbolic Cosecant, Hyperbolic Cosine, Hyperbolic Cotangent, Hyperbolic Functions,
Hyperbolic Secant, Hyperbolic Sine, Hyperbolic Tangent, Hyperelliptic Function, Hypergeometric
Function, Identity Function, Implicit Function, Implicit Function Theorem, Incomplete Gamma
Function, Increasing Function, Infinite Product, Instrument Function, Int, Inverse Cosecant,
Inverse Cosine, Inverse Cotangent, Inverse Function, Inverse Hyperbolic Functions, Inverse
Secant, Inverse Sine, Inverse Tangent, *j*-Function, Jacobi Elliptic Functions, Jacobi
Function of the First Kind, Jacobi Function of the Second Kind, Jacobi Theta Function, Jacobi Zeta
Function, Jinc Function, Joint Probability Density Function, Jonquière's Function, *K*-Function, Kei, Kelvin Functions, Ker, Koebe Function, *L*-Function,
Lambda Function, Lambda Hypergeometric Function, Lambert's *W*-Function, Lamé Function, Legendre Function of the First Kind, Legendre Function of the Second Kind, Lemniscate Function,
Lemniscate Function, Length Distribution Function, Lerch Transcendent, Lévy Function, Linearly Dependent Functions, Liouville Function, Lipschitz Function, Logarithm,
Logarithmically Convex Function, Logit Transformation, Lommel Function, Lyapunov Function,
MacRobert's *E*-Function, Mangoldt Function, Mathieu Function, Measurable Function, Meijer's
*G*-Function, Meromorphic, Mertens Function, Mertz Apodization Function, Mittag-Leffler Function,
Möbius Function, Möbius Periodic Function, Mock Theta
Function, Modified Bessel Function of the First Kind, Modified Bessel Function of the Second Kind, Modified
Spherical Bessel Function, Modified Struve Function, Modular Function, Modular Gamma Function,
Modular Lambda Function, Moment-Generating Function, Monogenic Function, Monotonic Function,
Mu Function, Multiplicative Function, Multivalued Function, Multivariate Function, Neumann
Function, Nint, Nu Function, Null Function, Numeric Function, Oblate Spheroidal Wave
Function, Odd Function, Omega Function, One-Way Function, Parabolic Cylinder Function,
Partition Function *P*, Partition Function *Q*,
Parzen Apodization Function, Pearson-Cunningham
Function, Pearson's Function, Periodic Function, Planck's Radiation Function, Plurisubharmonic
Function, Pochhammer Symbol, Poincaré-Fuchs-Klein Automorphic Function, Poisson-Charlier Function, Polygamma Function, Polygenic Function, Polylogarithm,
Positive Definite Function, Potential Function, Power, Prime Counting Function, Prime
Difference Function, Probability Density Function, Probability Distribution Function, Prolate Spheroidal
Wave Function, Psi Function, Pulse Function, *q*-Beta Function,
*Q*-Function, *q*-Gamma
Function, Quasiperiodic Function, Rademacher Function, Ramanujan Function, Ramanujan *g*- and *G*-
Functions, Ramanujan Theta Functions, Ramp Function, Rational Function,
Real Function, Rectangle Function, Regular Function, Regularized Gamma Function, Restricted
Divisor Function, Riemann Function, Riemann-Mangoldt Function, Riemann-Siegel Functions, Riemann
Theta Function, Riemann Zeta Function, Ring Function,
Sal, Sampling Function, Scalar Function, Schlomilch's Function, Secant, Sequency
Function, Sgn, Shah Function, Siegel Modular Function, Sigma Function, Sigmoid Function,
Sign, Sinc Function, Sine, Smarandache Function, Spence's Function, Spherical Bessel
Function of the First Kind, Spherical Bessel Function of the Second Kind, Spherical Hankel Function of the First
Kind, Spherical Hankel Function of the Second Kind, Spherical Harmonic, Spheroidal Wavefunction,
Sprague-Grundy Function, Square Root, Squared, Step Function, Struve Function, Sturm
Function, Summatory Function, Symmetric Function, TAK Function, Tangent, Tapering Function,
Tau Function, Tetrachoric Function, Theta Function, Toroidal Function, Toronto Function,
Total Function, Totient Function, Totient Valence Function, Transcendental Function, Transfer
Function, Trapdoor Function, Triangle Center Function, Triangle Function, Tricomi Function,
Trigonometric Functions, Uniform Apodization Function, Univalent Function, Vector Function,
Versine, von Mangoldt Function, *W*-Function, Walsh Function, Weber Functions,
Weierstraß Elliptic Function, Weierstraß Function, Weierstraß Sigma Function, Weierstraß Zeta
Function, Weighting Function, Welch Apodization Function, Whittaker Function,
Wiener Function, Window Function, Xi Function, Zeta Function

**References**

Functions

Abramowitz, M. and Stegun, C. A. (Eds.). ``Miscellaneous Functions.'' Ch. 27 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 997-1010, 1972.

Arfken, G. ``Special Functions.'' Ch. 13 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 712-759, 1985.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Special Functions.'' Ch. 6 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 205-265, 1992.

© 1996-9 *Eric W. Weisstein *

1999-05-26