List of mathematical functions

<h2 id="elementary-functions">Elementary functions</h2>
<p><a href="/facts/Elementary_functions/wVFyDwu8">Elementary functions</a> are functions built from basic operations (e.g. addition, exponentials, logarithms...)
</p>
<h3>Algebraic functions</h3>
<p><a href="/facts/Algebraic_function/yT7EKmwF">Algebraic functions</a> are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
</p>
<ul><li><a href="/facts/Polynomial/Lzak8VVx">Polynomials</a>: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
<ul><li><a href="/facts/Constant_function/BmPx6dn7">Constant function</a>: polynomial of degree zero, graph is a horizontal straight line</li>
<li><a href="/facts/Linear_function/Ejvy8CKy">Linear function</a>: First degree polynomial, graph is a straight line.</li>
<li><a href="/facts/Quadratic_function/mKxHIC3U">Quadratic function</a>: Second degree polynomial, graph is a <a href="/facts/Parabola/Cweo3wgw">parabola</a>.</li>
<li><a href="/facts/Cubic_function/t67bqD6c">Cubic function</a>: Third degree polynomial.</li>
<li><a href="/facts/Quartic_function/lHuBhMk7">Quartic function</a>: Fourth degree polynomial.</li>
<li><a href="/facts/Quintic_function/oeNDGj9l">Quintic function</a>: Fifth degree polynomial.</li></ul></li>
<li><a href="/facts/Rational_function/2nJGu0Ko">Rational functions</a>: A ratio of two polynomials.</li>
<li><a href="/facts/Nth_root/XrRjtJGe"><i>n</i>th root</a>
<ul><li><a href="/facts/Square_root/AfrzfBdQ">Square root</a>: Yields a number whose square is the given one.</li>
<li><a href="/facts/Cube_root/c8klHLFb">Cube root</a>: Yields a number whose cube is the given one.</li></ul></li></ul>
<h3>Elementary transcendental functions</h3>
<p><a href="/facts/Transcendental_function/yMkeq4Eo">Transcendental functions</a> are functions that are not algebraic.
</p>
<ul><li><a href="/facts/Exponential_function/ewlYxvR2">Exponential function</a>: raises a fixed number to a variable power.</li>
<li><a href="/facts/Hyperbolic_function/Y4Cb5S35">Hyperbolic functions</a>: formally similar to the <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a>.
<ul><li><a href="/facts/Inverse_hyperbolic_functions/myk0wJty">Inverse hyperbolic functions</a>: <a href="/facts/Inverse_function/Tg5nnXlu">inverses</a> of the <a href="/facts/Hyperbolic_functions/Y4Cb5S35">hyperbolic functions</a>, analogous to the <a href="/facts/Inverse_trigonometric_functions/bVwU9FPG">inverse circular functions</a>.</li></ul></li></ul>
<ul><li><a href="/facts/Logarithm/QeXaV97q">Logarithms</a>: the inverses of exponential functions; useful to solve equations involving exponentials.
<ul><li><a href="/facts/Natural_logarithm/PnreKEzI">Natural logarithm</a></li>
<li><a href="/facts/Common_logarithm/q6EhDnWk">Common logarithm</a></li>
<li><a href="/facts/Binary_logarithm/Uy7kAtRm">Binary logarithm</a></li></ul></li>
<li><a href="/facts/Exponentiation/pac6QY2g">Power functions</a>: raise a variable number to a fixed power; also known as <a href="/facts/Allometric_function/jERkArka">Allometric functions</a>; note: if the power is a rational number it is not strictly a transcendental function.</li>
<li><a href="/facts/Periodic_function/Sm8lJQsF">Periodic functions</a>
<ul><li><a href="/facts/Trigonometric_function/czej7ur0">Trigonometric functions</a>: <a href="/facts/Sine/gQ6LXK8z">sine</a>, <a href="/facts/Cosine/czej7ur0">cosine</a>, <a href="/facts/Tangent_(trigonometry)/czej7ur0">tangent</a>, <a href="/facts/Cotangent/czej7ur0">cotangent</a>, <a href="/facts/Secant_(trigonometry)/czej7ur0">secant</a>, <a href="/facts/Cosecant/czej7ur0">cosecant</a>, <a href="/facts/Exsecant/CNKrYMjC">exsecant</a>, <a href="/facts/Excosecant/CNKrYMjC">excosecant</a>, <a href="/facts/Versine/MllE4ISd">versine</a>, <a href="/facts/Coversine/MllE4ISd">coversine</a>, <a href="/facts/Vercosine/MllE4ISd">vercosine</a>, <a href="/facts/Covercosine/MllE4ISd">covercosine</a>, <a href="/facts/Haversine/MllE4ISd">haversine</a>, <a href="/facts/Hacoversine/MllE4ISd">hacoversine</a>, <a href="/facts/Havercosine/MllE4ISd">havercosine</a>, <a href="/facts/Hacovercosine/MllE4ISd">hacovercosine</a>, <a href="/facts/Inverse_trigonometric_functions/bVwU9FPG">Inverse trigonometric functions</a> etc.; used in <a href="/facts/Geometry/wTQf5ffQ">geometry</a> and to describe periodic phenomena. See also <a href="/facts/Gudermannian_function/bA0B2KcC">Gudermannian function</a>.</li></ul></li></ul>
<h2 id="special-functions">Special functions</h2>
<p class="note">Main article: <a href="/facts/Special_functions/JETSeucA">Special functions</a></p>
<h3>Piecewise special functions</h3>

<ul><li><a href="/facts/Indicator_function/QEuh04NM">Indicator function</a>: maps <i>x</i> to either 1 or 0, depending on whether or not <i>x</i> belongs to some subset.</li>
<li><a href="/facts/Step_function/VnrEjo4B">Step function</a>: A finite <a href="/facts/Linear_combination/CVjL6kuN">linear combination</a> of <a href="/facts/Indicator_function/QEuh04NM">indicator functions</a> of <a href="/facts/Half-open_interval/e9se3vTe">half-open intervals</a>.
<ul><li><a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>: 0 for negative arguments and 1 for positive arguments. The integral of the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>.</li></ul></li>
<li><a href="/facts/Sawtooth_wave/3QA8wW0h">Sawtooth wave</a></li>
<li><a href="/facts/Square_wave_(waveform)/h1cWR1KM">Square wave</a></li>
<li><a href="/facts/Triangle_wave/72BUtbRl">Triangle wave</a></li>
<li><a href="/facts/Rectangular_function/cVm8QG4F">Rectangular function</a></li>
<li><a href="/facts/Floor_function/ssehyMMf">Floor function</a>: Largest integer less than or equal to a given number.</li>
<li><a href="/facts/Ceiling_function/ssehyMMf">Ceiling function</a>: Smallest integer larger than or equal to a given number.</li>
<li><a href="/facts/Sign_function/VADy9zQq">Sign function</a>: Returns only the sign of a number, as +1, −1 or 0.</li>
<li><a href="/facts/Absolute_value/dwJBpEs5">Absolute value</a>: distance to the origin (zero point)</li></ul>
<h3>Arithmetic functions</h3>
<p class="note">Main article: <a href="/facts/Arithmetic_function/hULxwsgm">Arithmetic function</a></p>
<ul><li><a href="/facts/Divisor_function/rzvm1jw2">Sigma function</a>: <a href="/facts/Summation/V2WUxoKn">Sums</a> of <a href="/facts/Exponentiation/pac6QY2g">powers</a> of <a href="/facts/Divisor/DKNa2GvC">divisors</a> of a given <a href="/facts/Natural_number/u65og8JE">natural number</a>.</li>
<li><a href="/facts/Euler%2527s_totient_function/mIQwXYzj">Euler's totient function</a>: Number of numbers <a href="/facts/Coprime/bnCxCXxs">coprime</a> to (and not bigger than) a given one.</li>
<li><a href="/facts/Prime-counting_function/Dk1xJzsA">Prime-counting function</a>:  Number of <a href="/facts/Prime_number/L20FLkgn">primes</a> less than or equal to a given number.</li>
<li><a href="/facts/Partition_function_(number_theory)/mBUDy5vU">Partition function</a>:  Order-independent count of ways to write a given positive integer as a sum of positive integers.</li>
<li><a href="/facts/M%25C3%25B6bius_function/geuvVfqS">Möbius μ function</a>: Sum of the nth primitive roots of unity, it depends on the prime factorization of n.</li>
<li><a href="/facts/Prime_omega_function/obfFNVEB">Prime omega functions</a></li>
<li><a href="/facts/Chebyshev_function/Kufg2P2S">Chebyshev functions</a></li>
<li><a href="/facts/Liouville_function/HCP4Ukr5">Liouville function</a>, λ(<i>n</i>) = (–1)Ω(<i>n</i>)</li>
<li><a href="/facts/Von_Mangoldt_function/wwHKs2bg">Von Mangoldt function</a>, Λ(<i>n</i>) = log <i>p</i> if <i>n</i> is a positive power of the prime <i>p</i></li>
<li><a href="/facts/Carmichael_function/D0l0I6od">Carmichael function</a></li></ul>
<h3>Antiderivatives of elementary functions</h3>
<ul><li><a href="/facts/Logarithmic_integral_function/P3xwqRlA">Logarithmic integral function</a>: Integral of the reciprocal of the logarithm, important in the <a href="/facts/Prime_number_theorem/3KA9egls">prime number theorem</a>.</li>
<li><a href="/facts/Exponential_integral/JF6aMcmL">Exponential integral</a></li>
<li><a href="/facts/Trigonometric_integral/JB75EqnB">Trigonometric integral</a>: Including Sine Integral and Cosine Integral</li>
<li><a href="/facts/Inverse_tangent_integral/CKrkJMKn">Inverse tangent integral</a></li>
<li><a href="/facts/Error_function/Ca2H7pGr">Error function</a>: An integral important for <a href="/facts/Normal_distribution/UapjjPyQ">normal random variables</a>.
<ul><li><a href="/facts/Fresnel_integral/5W7z760j">Fresnel integral</a>: related to the error function; used in <a href="/facts/Optics/p0vq4S1T">optics</a>.</li>
<li><a href="/facts/Dawson_function/V89wVW9g">Dawson function</a>: occurs in <a href="/facts/Probability/fgYvMhwb">probability</a>.</li>
<li><a href="/facts/Faddeeva_function/NQ8ekfF2">Faddeeva function</a></li></ul></li></ul>
<h3>Gamma and related functions</h3>
<ul><li><a href="/facts/Gamma_function/SjGQcnyw">Gamma function</a>: A generalization of the <a href="/facts/Factorial/kkfy1Bwe">factorial</a> function.</li>
<li><a href="/facts/Barnes_G-function/4bmYxJrE">Barnes G-function</a></li>
<li><a href="/facts/Beta_function/Dcm1JAxb">Beta function</a>: Corresponding <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficient</a> analogue.</li>
<li><a href="/facts/Digamma_function/nDTNu86e">Digamma function</a>, <a href="/facts/Polygamma_function/IF6W68aL">Polygamma function</a></li>
<li><a href="/facts/Incomplete_beta_function/Dcm1JAxb">Incomplete beta function</a></li>
<li><a href="/facts/Incomplete_gamma_function/vojnYSu4">Incomplete gamma function</a></li>
<li><a href="/facts/K-function/XVxp0lgJ">K-function</a></li>
<li><a href="/facts/Multivariate_gamma_function/G8ATeYBq">Multivariate gamma function</a>: A generalization of the Gamma function useful in <a href="/facts/Multivariate_statistics/tDuNcWrL">multivariate statistics</a>.</li>
<li><a href="/facts/Student%2527s_t-distribution/DeT1SDqH">Student's t-distribution</a></li>
<li><a href="/facts/Gamma_function/SjGQcnyw">Pi function</a> 
  
    
      
        Π
        (
        z
        )
        =
        z
        Γ
        (
        z
        )
        =
        (
        z
        )
        !
      
    
    {\displaystyle \Pi (z)=z\Gamma (z)=(z)!}
  
</li></ul>
<h3>Elliptic and related functions</h3>

<ul><li><a href="/facts/Elliptic_integral/q8EGYzQ3">Elliptic integrals</a>: Arising from the path length of <a href="/facts/Ellipse/1wUDnGxY">ellipses</a>; important in many applications. Alternate notations include:
<ul><li><a href="/facts/Carlson_symmetric_form/w1W9gCqr">Carlson symmetric form</a></li>
<li><a href="/facts/Legendre_form/rHCMxbmH">Legendre form</a></li></ul></li>
<li><a href="/facts/Nome_(mathematics)/C3yIM7eR">Nome</a></li>
<li><a href="/facts/Quarter_period/BeQx1WRb">Quarter period</a></li>
<li><a href="/facts/Elliptic_function/mF03oI04">Elliptic functions</a>: The inverses of elliptic integrals; used to model double-periodic phenomena.
<ul><li><a href="/facts/Jacobi%2527s_elliptic_functions/BLk6Rxew">Jacobi's elliptic functions</a></li>
<li><a href="/facts/Weierstrass%2527s_elliptic_functions/GHRXDknk">Weierstrass's elliptic functions</a></li>
<li><a href="/facts/Lemniscate_elliptic_functions/cHWb3moy">Lemniscate elliptic functions</a></li></ul></li>
<li><a href="/facts/Theta_functions/g09xvoQQ">Theta functions</a></li>
<li><a href="/facts/Neville_theta_functions/Rv5PW661">Neville theta functions</a></li>
<li><a href="/facts/Modular_lambda_function/k30w0QWf">Modular lambda function</a></li>
<li>Closely related are the <a href="/facts/Modular_form/PcnbYR1f">modular forms</a>, which include
<ul><li><a href="/facts/J-invariant/H1a38AJF">J-invariant</a></li>
<li><a href="/facts/Dedekind_eta_function/nStBqqmo">Dedekind eta function</a></li></ul></li></ul>
<h3>Bessel and related functions</h3>

<ul><li><a href="/facts/Airy_function/bydfB54Y">Airy function</a></li>
<li><a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a>: Defined by a <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a>; useful in <a href="/facts/Astronomy/0ArJVM1p">astronomy</a>, <a href="/facts/Electromagnetism/oAkmALHT">electromagnetism</a>, and <a href="/facts/Mechanics/A3yHLQA5">mechanics</a>.</li>
<li><a href="/facts/Bessel%25E2%2580%2593Clifford_function/BWX2owAe">Bessel–Clifford function</a></li>
<li><a href="/facts/Kelvin_functions/EN3suFCm">Kelvin functions</a></li>
<li><a href="/facts/Legendre_function/sDo3ra2G">Legendre function</a>: From the theory of <a href="/facts/Spherical_harmonics/bn98Gv2L">spherical harmonics</a>.</li>
<li><a href="/facts/Scorer%2527s_function/nEFHIbao">Scorer's function</a></li>
<li><a href="/facts/Sinc_function/lSvDtHy0">Sinc function</a></li>
<li><a href="/facts/Hermite_polynomials/MLQNY0ez">Hermite polynomials</a></li>
<li><a href="/facts/Laguerre_polynomials/7RM3ypWl">Laguerre polynomials</a></li>
<li><a href="/facts/Chebyshev_polynomials/HoBINqgC">Chebyshev polynomials</a></li>
<li><a href="/facts/Synchrotron_function/RME40hJQ">Synchrotron function</a></li></ul>
<h3>Riemann zeta and related functions</h3>

<ul><li><a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>: A special case of <a href="/facts/Dirichlet_series/uIAUaLcd">Dirichlet series</a>.</li>
<li><a href="/facts/Riemann_Xi_function/Uz0j5Kkr">Riemann Xi function</a></li>
<li><a href="/facts/Dirichlet_eta_function/32tUb6a7">Dirichlet eta function</a>: An allied function.</li>
<li><a href="/facts/Dirichlet_beta_function/FKwkVguL">Dirichlet beta function</a></li>
<li><a href="/facts/Dirichlet_L-function/my8hc6Qy">Dirichlet L-function</a></li>
<li><a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a></li>
<li><a href="/facts/Legendre_chi_function/QF7pSJNt">Legendre chi function</a></li>
<li><a href="/facts/Lerch_transcendent/26RvsFok">Lerch transcendent</a></li>
<li><a href="/facts/Polylogarithm/CKrkJMKn">Polylogarithm</a> and related functions:
<ul><li><a href="/facts/Incomplete_polylogarithm/IlpxhNQ6">Incomplete polylogarithm</a></li>
<li><a href="/facts/Clausen_function/1mDkvwcL">Clausen function</a></li>
<li><a href="/facts/Complete_Fermi%25E2%2580%2593Dirac_integral/0Ez2xIkd">Complete Fermi–Dirac integral</a>, an alternate form of the polylogarithm.</li>
<li><a href="/facts/Dilogarithm/lsnCj3Gb">Dilogarithm</a></li>
<li><a href="/facts/Incomplete_Fermi%25E2%2580%2593Dirac_integral/5MdFPKII">Incomplete Fermi–Dirac integral</a></li>
<li><a href="/facts/Kummer%2527s_function/hmNeJ0GD">Kummer's function</a></li></ul></li>
<li><a href="/facts/Riesz_function/4IuklIdV">Riesz function</a></li></ul>
<h3>Hypergeometric and related functions</h3>
<ul><li><a href="/facts/Hypergeometric_function/yTEC6NBJ">Hypergeometric functions</a>: Versatile family of <a href="/facts/Power_series/vYh6sTps">power series</a>.</li>
<li><a href="/facts/Confluent_hypergeometric_function/D0FxgWUa">Confluent hypergeometric function</a></li>
<li><a href="/facts/Associated_Legendre_functions/sDo3ra2G">Associated Legendre functions</a></li>
<li><a href="/facts/Meijer_G-function/YCxbgSAl">Meijer G-function</a></li>
<li><a href="/facts/Fox_H-function/QwfvM8IN">Fox H-function</a></li></ul>
<h3>Iterated exponential and related functions</h3>
<ul><li><a href="/facts/Hyper_operator/XNn1hneN">Hyper operators</a></li>
<li><a href="/facts/Iterated_logarithm/bEY9cfh8">Iterated logarithm</a></li>
<li><a href="/facts/Pentation/YKhnIWPG">Pentation</a></li>
<li><a href="/facts/Super-logarithm/2SeXkylX">Super-logarithms</a></li>
<li><a href="/facts/Tetration/5rKeVVYv">Tetration</a></li></ul>
<h3>Other standard special functions</h3>
<ul><li><a href="/facts/Lambert_W_function/NG1wd71f">Lambert W function</a>: Inverse of <i>f</i>(<i>w</i>) = <i>w</i> exp(<i>w</i>).</li>
<li><a href="/facts/Lam%25C3%25A9_function/UlsAqXlS">Lamé function</a></li>
<li><a href="/facts/Mathieu_function/zmsyWw0K">Mathieu function</a></li>
<li><a href="/facts/Mittag-Leffler_function/qWjX1gP2">Mittag-Leffler function</a></li>
<li><a href="/facts/Painlev%25C3%25A9_transcendents/wIkH4Y1G">Painlevé transcendents</a></li>
<li><a href="/facts/Parabolic_cylinder_function/YhD8JEfy">Parabolic cylinder function</a></li>
<li><a href="/facts/Arithmetic%25E2%2580%2593geometric_mean/8PRN9UB4">Arithmetic–geometric mean</a></li></ul>
<h3>Miscellaneous functions</h3>
<ul><li><a href="/facts/Ackermann_function/WQNpCWHn">Ackermann function</a>: in the <a href="/facts/Theory_of_computation/xYJd2YeI">theory of computation</a>, a <a href="/facts/Computable_function/dzwBlLGn">computable function</a> that is not <a href="/facts/Primitive_recursive_function/4z97gRnl">primitive recursive</a>.</li>
<li><a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>: everywhere zero except for <i>x</i> = 0; total integral is 1. Not a function but a <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distribution</a>, but sometimes informally referred to as a function, particularly by physicists and engineers.</li>
<li><a href="/facts/Dirichlet_function/d9o708HR">Dirichlet function</a>: is an <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> that matches 1 to <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> and 0 to <a href="/facts/Irrational_number/Psv19TET">irrationals</a>. It is <a href="/facts/Nowhere_continuous/XbxlDzx1">nowhere continuous</a>.</li>
<li><a href="/facts/Thomae%2527s_function/SjCt2ahN">Thomae's function</a>: 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.</li>
<li><a href="/facts/Kronecker_delta_function/gcGAy0bm">Kronecker delta function</a>: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.</li>
<li><a href="/facts/Minkowski%2527s_question_mark_function/gum3I0pX">Minkowski's question mark function</a>: Derivatives vanish on the rationals.</li>
<li><a href="/facts/Weierstrass_function/m9jU4LE5">Weierstrass function</a>: is an example of <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> that is nowhere <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_types_of_functions/9wje5aIS">List of types of functions</a></li>
<li><a href="/facts/Test_functions_for_optimization/lSafL6bc">Test functions for optimization</a></li>
<li><a href="/facts/List_of_mathematical_abbreviations/yWsqSmQv">List of mathematical abbreviations</a></li>
<li><a href="/facts/List_of_special_functions_and_eponyms/SgVtbePW">List of special functions and eponyms</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://archive.today/20130105073730/http://www.special-functions.com/">Special functions</a> : A programmable special functions calculator.</li>
<li><a href="http://eqworld.ipmnet.ru/en/auxiliary/aux-specfunc.htm">Special functions</a> at EqWorld: The World of Mathematical Equations.</li></ul>

List of mathematical functions open-in-new

List of mathematical functions