{"history":[],"article":{"id":"176ff587-33cc-4a7f-a746-a496d6e897d4","title":"Error function","slug":"error-function","content":":::summary\n![Plot of the error function over real numbers](https://upload.wikimedia.org/wikipedia/commons/2/2f/Error///Function.svg)\n**Name:** Error function \n**Imagesize:** 400px \n**Imagealt:** Plot of the error function over real numbers \n**Fields Of Application:** Probability, thermodynamics, digital communications \n**Domain:** \\mathbb{R} \n**Parity:** Odd \n**Root:** 0 \n:::\n\n# \nIn mathematics, the **error function** (also called the Gauss **error function**), often denoted by '''\nerf\n''', is a function defined as:\nThe integral here is a complex [contour integral](https://bliptext.com/articles/contour-integration) which is path-independent because \\exp(-t^2) is [holomorphic](https://bliptext.com/articles/holomorphic-function) on the whole complex plane \\mathbb{C}.\nIn many applications, the function argument is a real number, in which case the function value is also real.\nIn some old texts,\nthe error function is defined without the factor of.\nThis [nonelementary integral](https://bliptext.com/articles/nonelementary-integral) is a [sigmoid](https://bliptext.com/articles/sigmoid-function) function that occurs often in [probability](https://bliptext.com/articles/probability), [statistics](https://bliptext.com/articles/statistics), and [partial differential equations](https://bliptext.com/articles/partial-differential-equation).\nIn statistics, for non-negative real values of x, the error function has the following interpretation: for a real [random variable](https://bliptext.com/articles/random-variable) Y that is [normally distributed](https://bliptext.com/articles/normal-distribution) with [mean](https://bliptext.com/articles/mean) 0 and [standard deviation](https://bliptext.com/articles/standard-deviation) 1/\\sqrt{2},\nerf x\nis the probability that Y falls in the range.\nTwo closely related functions are the **complementary error function** is defined as\nand the **imaginary error function** is defined as\nwhere i is the [imaginary unit](https://bliptext.com/articles/imaginary-unit).\n\n# Name\nThe name \"error function\" and its abbreviation\nerf\nwere proposed by [J. W. L. Glaisher](https://bliptext.com/articles/james-whitbread-lee-glaisher) in 1871 on account of its connection with \"the theory of Probability, and notably the theory of [Errors](https://bliptext.com/articles/errors-and-residuals).\"\nThe error function complement was also discussed by Glaisher in a separate publication in the same year.\nFor the \"law of facility\" of errors whose [density](https://bliptext.com/articles/probability-density) is given by\n(the [normal distribution](https://bliptext.com/articles/normal-distribution)), Glaisher calculates the probability of an error lying between p and q as:\n# [Plot of the error function Erf(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D | upload.wikimedia.org/wikipedia/commons/9/92/Plot///of///the///error///function///Erf%28z%29///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///created///with///Mathematica///13.1///function///ComplexPlot3D.svg]\n\n# Applications\nWhen the results of a series of measurements are described by a [normal distribution](https://bliptext.com/articles/normal-distribution) with [standard deviation](https://bliptext.com/articles/standard-deviation) σ and [expected value](https://bliptext.com/articles/expected-value) 0, then\nerf (a⁄σ √2)\nis the probability that the error of a single measurement lies between\n−a\nand\n+a\n, for positive a. This is useful, for example, in determining the [bit error rate](https://bliptext.com/articles/bit-error-rate) of a digital communication system.\nThe error and complementary error functions occur, for example, in solutions of the [heat equation](https://bliptext.com/articles/heat-equation) when [boundary conditions](https://bliptext.com/articles/boundary-condition) are given by the [Heaviside step function](https://bliptext.com/articles/heaviside-step-function).\nThe error function and its approximations can be used to estimate results that hold [with high probability](https://bliptext.com/articles/with-high-probability) or with low probability.\nGiven a random variable\nX ~ Norm[μ,σ]\n(a normal distribution with mean μ and standard deviation σ) and a constant\nL > μ\n, it can be shown via integration by substitution:\nwhere A and B are certain numeric constants.\nIf L is sufficiently far from the mean, specifically\nμ − L ≥ σ√ln k\n, then:\nso the probability goes to 0 as\nk → ∞\n.\nThe probability for X being in the interval can be derived as\n\n# Properties\nThe property\nerf (−z) = −erf z\nmeans that the error function is an [odd function](https://bliptext.com/articles/even-and-odd-functions).\nThis directly results from the fact that the integrand\ne−t 2\nis an [even function](https://bliptext.com/articles/even-function) (the antiderivative of an [even function](https://bliptext.com/articles/even-function) which is zero at the origin is an odd function and vice versa).\nSince the error function is an [entire function](https://bliptext.com/articles/entire-function) which takes real numbers to real numbers, for any [complex number](https://bliptext.com/articles/complex-number) z:\nwhere\nz\nis the [complex conjugate](https://bliptext.com/articles/complex-conjugate) of z.\nThe integrand\nf = exp(−z2)\nand\nf = erf z\nare shown in the complex z-plane in the figures at right with [domain coloring](https://bliptext.com/articles/domain-coloring).\nThe error function at\n+∞\nis exactly 1 (see [Gaussian integral](https://bliptext.com/articles/gaussian-integral)).\nAt the real axis,\nerf z\napproaches unity at\nz → +∞\nand −1 at\nz → −∞\n. At the imaginary axis, it tends to\n±i∞\n.\n\n## Taylor series\nThe error function is an [entire function](https://bliptext.com/articles/entire-function); it has no singularities (except that at infinity) and its [Taylor expansion](https://bliptext.com/articles/taylor-expansion) always converges.\nFor\nx >> 1\n, however, cancellation of leading terms makes the Taylor expansion unpractical.\nThe defining integral cannot be evaluated in [closed form](https://bliptext.com/articles/closed-form-expression) in terms of [elementary functions](https://bliptext.com/articles/elementary-function-differential-algebra) (see [Liouville's theorem](https://bliptext.com/articles/liouville-s-theorem-differential-algebra)), but by expanding the [integrand](https://bliptext.com/articles/integrand)\ne−z 2\ninto its [Maclaurin series](https://bliptext.com/articles/maclaurin-series) and integrating term by term, one obtains the error function's [Maclaurin series](https://bliptext.com/articles/maclaurin-series) as:\nwhich holds for every [complex number](https://bliptext.com/articles/complex-number) z. The denominator terms are sequence [A007680](https://bliptext.com/articles/a007680) in the [OEIS](https://bliptext.com/articles/oeis).\nFor iterative calculation of the above series, the following alternative formulation may be useful:\nbecause\n−(2k − 1)z2⁄k(2k + 1)\nexpresses the multiplier to turn the kth term into the\n(k + 1)\nth term (considering z as the first term).\nThe imaginary error function has a very similar Maclaurin series, which is:\nwhich holds for every [complex number](https://bliptext.com/articles/complex-number) z.\n\n## Derivative and integral\nThe derivative of the error function follows immediately from its definition:\nFrom this, the derivative of the imaginary error function is also immediate:\nAn [antiderivative](https://bliptext.com/articles/antiderivative) of the error function, obtainable by [integration by parts](https://bliptext.com/articles/integration-by-parts), is\nAn antiderivative of the imaginary error function, also obtainable by integration by parts, is\nHigher order derivatives are given by\nwhere H are the physicists' [Hermite polynomials](https://bliptext.com/articles/hermite-polynomials).\n\n## Bürmann series\nAn expansion, which converges more rapidly for all real values of x than a Taylor expansion, is obtained by using [Hans Heinrich Bürmann](https://bliptext.com/articles/hans-heinrich-b-rmann)'s theorem:\nwhere\nsgn\nis the [sign function](https://bliptext.com/articles/sign-function).\nBy keeping only the first two coefficients and choosing\nc1 = 31⁄200\nand\nc2 = −341⁄8000\n, the resulting approximation shows its largest relative error at\nx = ±1.3796\n, where it is less than 0.0036127:\n\n## Inverse functions\nGiven a complex number z, there is not a unique complex number w satisfying\nerf w = z\n, so a true inverse function would be multivalued.\nHowever, for\n−1 < x < 1\n, there is a unique real number denoted\nerf−1 x\nsatisfying\nThe **inverse error function** is usually defined with domain, and it is restricted to this domain in many computer algebra systems.\nHowever, it can be extended to the disk\nof the complex plane, using the Maclaurin series\nwhere\nc0 = 1\nand\nSo we have the series expansion (common factors have been canceled from numerators and denominators):\n(After cancellation the numerator and denominator values in and respectively; without cancellation the numerator terms are values in .) The error function's value at\n±∞\nis equal to\n±1\n.\nFor\n, we have\nerf(erf−1 z) = z\n.\nThe **inverse complementary error function** is defined as\nFor real x, there is a unique real number\nerfi−1 x\nsatisfying\nerfi(erfi−1 x) = x\n. The **inverse imaginary error function** is defined as\nerfi−1 x\n.\nFor any real x, [Newton's method](https://bliptext.com/articles/newton-s-method) can be used to compute\nerfi−1 x\n, and for\n−1 ≤ x ≤ 1\n, the following Maclaurin series converges:\nwhere\nck\nis defined as above.\n# [Inverse error function | upload.wikimedia.org/wikipedia/commons/e/ec/Mplwp///erf///inv.svg]\n\n## Asymptotic expansion\nA useful [asymptotic expansion](https://bliptext.com/articles/asymptotic-expansion) of the complementary error function (and therefore also of the error function) for large real x is\nwhere\n(2n − 1)!!\nis the [double factorial](https://bliptext.com/articles/double-factorial) of\n(2n − 1)\n, which is the product of all odd numbers up to\n(2n − 1)\n. This series diverges for every finite x, and its meaning as asymptotic expansion is that for any integer\nN ≥ 1\none has\nwhere the remainder is\nwhich follows easily by induction, writing\nand integrating by parts.\nThe asymptotic behavior of the remainder term, in [Landau notation](https://bliptext.com/articles/landau-notation), is\nas\nx → ∞\n. This can be found by\nFor large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of\nerfc x\n(while for not too large values of x, the above Taylor expansion at 0 provides a very fast convergence).\n\n## Continued fraction expansion\nA [continued fraction](https://bliptext.com/articles/continued-fraction) expansion of the complementary error function was found by [Laplace](https://bliptext.com/articles/pierre-simon-laplace):\n\n## Factorial series\nThe inverse [factorial series](https://bliptext.com/articles/factorial-series):\nconverges for\nRe(z2) > 0\n. Here\nzn\ndenotes the [rising factorial](https://bliptext.com/articles/rising-factorial), and\ns(n,k)\ndenotes a signed [Stirling number of the first kind](https://bliptext.com/articles/stirling-number-of-the-first-kind).\nThere also exists a representation by an infinite sum containing the [double factorial](https://bliptext.com/articles/double-factorial):\n\n# Numerical approximations\n\n## Approximation with elementary functions\n

\n[Abramowitz and Stegun](https://bliptext.com/articles/abramowitz-and-stegun) give several approximations of varying accuracy (equations 7.1.25–28).\nThis allows one to choose the fastest approximation suitable for a given application.\nIn order of increasing accuracy, they are:\n(maximum error: 0.001)\nwhere\n,\n,\n,\n(maximum error: 0)\nwhere\n,\n,\n,\n(maximum error: 0)\nwhere\n,\n,\n,\n,\n,\n(maximum error: 0)\nwhere\n,\n,\n,\n,\n,\nAll of these approximations are valid for\nx ≥ 0\n. To use these approximations for negative x, use the fact that\nerf x\nis an odd function, so\n.
\nExponential bounds and a pure exponential approximation for the complementary error function are given by\n
\nThe above have been generalized to sums of N exponentials with increasing accuracy in terms of N so that\nerfc x\ncan be accurately approximated or bounded by\n2Q̃(√2x)\n, where\nIn particular, there is a systematic methodology to solve the numerical coefficients\n{(an,bn)}N n = 1\nthat yield a [minimax](https://bliptext.com/articles/minimax-approximation-algorithm) approximation or bound for the closely related [Q-function](https://bliptext.com/articles/q-function):\nQ(x) ≈ Q̃(x)\n,\nQ(x) ≤ Q̃(x)\n, or\nQ(x) ≥ Q̃(x)\nfor\nx ≥ 0\n. The coefficients\n{(an,bn)}N n = 1\nfor many variations of the exponential approximations and bounds up to\nhave been released to open access as a comprehensive dataset.\n
\nA tight approximation of the complementary error function for\nx ∈ [0,∞)\nis given by [Karagiannidis](https://bliptext.com/articles/george-karagiannidis) & Lioumpas (2007) who showed for the appropriate choice of parameters\n{A,B}\nthat\nThey determined\n{A,B} = {1.98,1.135}\n, which gave a good approximation for all\nx ≥ 0\n. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.\n
\nA single-term lower bound is\nwhere the parameter β can be picked to minimize error on the desired interval of approximation.\n
\nAnother approximation is given by Sergei Winitzki using his \"global Padé approximations\":\nwhere\nThis is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the relative error is less than 0.00035 for all real x. Using the alternate value\na ≈ 0.147\nreduces the maximum relative error to about 0.00013.\nThis approximation can be inverted to obtain an approximation for the inverse error function:\n
\nAn approximation with a maximal error of 0 for any real argument is:\nwith\nand\n
An approximation of with a maximum relative error less than 2^{-53} in absolute value is:\nfor x\\ge 0,\nand for x<0\n
\nA simple approximation for real-valued arguments could be done through [Hyperbolic functions](https://bliptext.com/articles/hyperbolic-functions):\nwhich keeps the absolute difference.\n

\n\n## Table of values\n\n# Related functions\n\n## Complementary error function\nThe **complementary error function**, denoted\nerfc\n, is defined as\nwhich also defines\nerfcx\n, the **scaled complementary error function** (which can be used instead of\nerfc\nto avoid [arithmetic underflow](https://bliptext.com/articles/arithmetic-underflow) ). Another form of\nerfc x\nfor\nx ≥ 0\nis known as Craig's formula, after its discoverer:\nThis expression is valid only for positive values of x, but it can be used in conjunction with\nto obtain\nerfc(x)\nfor negative values.\nThis form is advantageous in that the range of integration is fixed and finite.\nAn extension of this expression for the\nerfc\nof the sum of two non-negative variables is as follows:\n# [Plot of the complementary error function Erfc(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D | upload.wikimedia.org/wikipedia/commons/4/46/Plot///of///the///complementary///error///function///Erfc%28z%29///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///created///with///Mathematica///13.1///function///ComplexPlot3D.svg]\n\n## Imaginary error function\nThe **imaginary error function**, denoted\nerfi\n, is defined as\nwhere\nD(x)\nis the [Dawson function](https://bliptext.com/articles/dawson-function) (which can be used instead of\nerfi\nto avoid [arithmetic overflow](https://bliptext.com/articles/arithmetic-overflow) ).\nDespite the name \"imaginary error function\",\nerfi x\nis real when x is real.\nWhen the error function is evaluated for arbitrary [complex](https://bliptext.com/articles/complex-number) arguments z, the resulting **[complex](https://bliptext.com/articles/complex-number) error function** is usually discussed in scaled form as the [Faddeeva function](https://bliptext.com/articles/faddeeva-function):\n# [Plot of the imaginary error function Erfi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D | upload.wikimedia.org/wikipedia/commons/b/b5/Plot///of///the///imaginary///error///function///Erfi%28z%29///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///created///with///Mathematica///13.1///function///ComplexPlot3D.svg]\n\n## Cumulative distribution function\nThe error function is essentially identical to the standard [normal cumulative distribution function](https://bliptext.com/articles/normal-cumulative-distribution-function), denoted\nΦ\n, also named\nnorm(x)\nby some software languages, as they differ only by scaling and translation.\nIndeed,\nor rearranged for\nerf\nand\nerfc\nConsequently, the error function is also closely related to the [Q-function](https://bliptext.com/articles/q-function), which is the tail probability of the standard normal distribution.\nThe Q-function can be expressed in terms of the error function as\nThe [inverse](https://bliptext.com/articles/inverse-function) of\nΦ\nis known as the [normal quantile function](https://bliptext.com/articles/quantile-function), or [probit](https://bliptext.com/articles/probit) function and may be expressed in terms of the inverse error function as\nThe standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.\nThe error function is a special case of the [Mittag-Leffler function](https://bliptext.com/articles/mittag-leffler-function), and can also be expressed as a [confluent hypergeometric function](https://bliptext.com/articles/confluent-hypergeometric-function) (Kummer's function):\nIt has a simple expression in terms of the [Fresnel integral](https://bliptext.com/articles/fresnel-integral).\nIn terms of the [regularized gamma function](https://bliptext.com/articles/regularized-gamma-function) P and the [incomplete gamma function](https://bliptext.com/articles/incomplete-gamma-function),\nsgn x\nis the [sign function](https://bliptext.com/articles/sign-function).\n# [the normal cumulative distribution function plotted in the complex plane | upload.wikimedia.org/wikipedia/commons/a/ad/Normal///cumulative///distribution///function///complex///plot///in///Mathematica///13.1///with///ComplexPlot3D.svg]\n\n## Iterated integrals of the complementary error function\nThe iterated integrals of the complementary error function are defined by\nThe general recurrence formula is\nThey have the power series\nfrom which follow the symmetry properties\nand\n\n# Implementations\n\n## As real function of a real argument\n\n## As complex function of a complex argument","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:33:04.518Z","updated_at":"2025-01-25T18:33:04.518Z","search_vector":"'error':1 'function':2","search_content":" \n \n**Name:** Error function \n**Imagesize:** 400px \n**Imagealt:** Plot of the error function over real numbers \n**Fields Of Application:** Probability, thermodynamics, digital communications \n**Domain:** \\mathbb{R} \n**Parity:** Odd \n**Root:** 0 \n \n\n# \nIn mathematics, the **error function** (also called the Gauss **error function**), often denoted by '''\nerf\n''', is a function defined as:\nThe integral here is a complex [contour integral](https://bliptext.com/articles/contour-integration) which is path-independent because \\exp(-t^2) is [holomorphic](https://bliptext.com/articles/holomorphic-function) on the whole complex plane \\mathbb{C}.\nIn many applications, the function argument is a real number, in which case the function value is also real.\nIn some old texts,\nthe error function is defined without the factor of.\nThis [nonelementary integral](https://bliptext.com/articles/nonelementary-integral) is a [sigmoid](https://bliptext.com/articles/sigmoid-function) function that occurs often in [probability](https://bliptext.com/articles/probability), [statistics](https://bliptext.com/articles/statistics), and [partial differential equations](https://bliptext.com/articles/partial-differential-equation).\nIn statistics, for non-negative real values of x, the error function has the following interpretation: for a real [random variable](https://bliptext.com/articles/random-variable) Y that is [normally distributed](https://bliptext.com/articles/normal-distribution) with [mean](https://bliptext.com/articles/mean) 0 and [standard deviation](https://bliptext.com/articles/standard-deviation) 1/\\sqrt{2},\nerf x\nis the probability that Y falls in the range.\nTwo closely related functions are the **complementary error function** is defined as\nand the **imaginary error function** is defined as\nwhere i is the [imaginary unit](https://bliptext.com/articles/imaginary-unit).\n\n# Name\nThe name \"error function\" and its abbreviation\nerf\nwere proposed by [J. W. L. Glaisher](https://bliptext.com/articles/james-whitbread-lee-glaisher) in 1871 on account of its connection with \"the theory of Probability, and notably the theory of [Errors](https://bliptext.com/articles/errors-and-residuals).\"\nThe error function complement was also discussed by Glaisher in a separate publication in the same year.\nFor the \"law of facility\" of errors whose [density](https://bliptext.com/articles/probability-density) is given by\n(the [normal distribution](https://bliptext.com/articles/normal-distribution)), Glaisher calculates the probability of an error lying between p and q as:\n# [Plot of the error function Erf(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D | upload.wikimedia.org/wikipedia/commons/9/92/Plot///of///the///error///function///Erf%28z%29///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///created///with///Mathematica///1 3.1///function///ComplexPlot3D.svg]\n\n# Applications\nWhen the results of a series of measurements are described by a [normal distribution](https://bliptext.com/articles/normal-distribution) with [standard deviation](https://bliptext.com/articles/standard-deviation) σ and [expected value](https://bliptext.com/articles/expected-value) 0, then\nerf (a⁄σ √2)\nis the probability that the error of a single measurement lies between\n−a\nand\n+a\n, for positive a. This is useful, for example, in determining the [bit error rate](https://bliptext.com/articles/bit-error-rate) of a digital communication system.\nThe error and complementary error functions occur, for example, in solutions of the [heat equation](https://bliptext.com/articles/heat-equation) when [boundary conditions](https://bliptext.com/articles/boundary-condition) are given by the [Heaviside step function](https://bliptext.com/articles/heaviside-step-function).\nThe error function and its approximations can be used to estimate results that hold [with high probability](https://bliptext.com/articles/with-high-probability) or with low probability.\nGiven a random variable\nX ~ Norm[μ,σ]\n(a normal distribution with mean μ and standard deviation σ) and a constant\nL > μ\n, it can be shown via integration by substitution:\nwhere A and B are certain numeric constants.\nIf L is sufficiently far from the mean, specifically\nμ − L ≥ σ√ln k\n, then:\nso the probability goes to 0 as\nk → ∞\n.\nThe probability for X being in the interval can be derived as\n\n# Properties\nThe property\nerf (−z) = −erf z\nmeans that the error function is an [odd function](https://bliptext.com/articles/even-and-odd-functions).\nThis directly results from the fact that the integrand\ne−t 2\nis an [even function](https://bliptext.com/articles/even-function) (the antiderivative of an [even function](https://bliptext.com/articles/even-function) which is zero at the origin is an odd function and vice versa).\nSince the error function is an [entire function](https://bliptext.com/articles/entire-function) which takes real numbers to real numbers, for any [complex number](https://bliptext.com/articles/complex-number) z:\nwhere\nz\nis the [complex conjugate](https://bliptext.com/articles/complex-conjugate) of z.\nThe integrand\nf = exp(−z2)\nand\nf = erf z\nare shown in the complex z-plane in the figures at right with [domain coloring](https://bliptext.com/articles/domain-coloring).\nThe error function at\n+∞\nis exactly 1 (see [Gaussian integral](https://bliptext.com/articles/gaussian-integral)).\nAt the real axis,\nerf z\napproaches unity at\nz → +∞\nand −1 at\nz → −∞\n. At the imaginary axis, it tends to\n±i∞\n.\n\n## Taylor series\nThe error function is an [entire function](https://bliptext.com/articles/entire-function); it has no singularities (except that at infinity) and its [Taylor expansion](https://bliptext.com/articles/taylor-expansion) always converges.\nFor\nx >> 1\n, however, cancellation of leading terms makes the Taylor expansion unpractical.\nThe defining integral cannot be evaluated in [closed form](https://bliptext.com/articles/closed-form-expression) in terms of [elementary functions](https://bliptext.com/articles/elementary-function-differential-algebra) (see [Liouville's theorem](https://bliptext.com/articles/liouville-s-theorem-differential-algebra)), but by expanding the [integrand](https://bliptext.com/articles/integrand)\ne−z 2\ninto its [Maclaurin series](https://bliptext.com/articles/maclaurin-series) and integrating term by term, one obtains the error function's [Maclaurin series](https://bliptext.com/articles/maclaurin-series) as:\nwhich holds for every [complex number](https://bliptext.com/articles/complex-number) z. The denominator terms are sequence [A007680](https://bliptext.com/articles/a007680) in the [OEIS](https://bliptext.com/articles/oeis).\nFor iterative calculation of the above series, the following alternative formulation may be useful:\nbecause\n−(2k − 1)z2⁄k(2k + 1)\nexpresses the multiplier to turn the kth term into the\n(k + 1)\nth term (considering z as the first term).\nThe imaginary error function has a very similar Maclaurin series, which is:\nwhich holds for every [complex number](https://bliptext.com/articles/complex-number) z.\n\n## Derivative and integral\nThe derivative of the error function follows immediately from its definition:\nFrom this, the derivative of the imaginary error function is also immediate:\nAn [antiderivative](https://bliptext.com/articles/antiderivative) of the error function, obtainable by [integration by parts](https://bliptext.com/articles/integration-by-parts), is\nAn antiderivative of the imaginary error function, also obtainable by integration by parts, is\nHigher order derivatives are given by\nwhere H are the physicists' [Hermite polynomials](https://bliptext.com/articles/hermite-polynomials).\n\n## Bürmann series\nAn expansion, which converges more rapidly for all real values of x than a Taylor expansion, is obtained by using [Hans Heinrich Bürmann](https://bliptext.com/articles/hans-heinrich-b-rmann)'s theorem:\nwhere\nsgn\nis the [sign function](https://bliptext.com/articles/sign-function).\nBy keeping only the first two coefficients and choosing\nc1 = 31⁄200\nand\nc2 = −341⁄8000\n, the resulting approximation shows its largest relative error at\nx = ±1.3796\n, where it is less than 0.0036127:\n\n## Inverse functions\nGiven a complex number z, there is not a unique complex number w satisfying\nerf w = z\n, so a true inverse function would be multivalued.\nHowever, for\n−1 0\n. Here\nzn\ndenotes the [rising factorial](https://bliptext.com/articles/rising-factorial), and\ns(n,k)\ndenotes a signed [Stirling number of the first kind](https://bliptext.com/articles/stirling-number-of-the-first-kind).\nThere also exists a representation by an infinite sum containing the [double factorial](https://bliptext.com/articles/double-factorial):\n\n# Numerical approximations\n\n## Approximation with elementary functions\n \n \n[Abramowitz and Stegun](https://bliptext.com/articles/abramowitz-and-stegun) give several approximations of varying accuracy (equations 7.1.25–28).\nThis allows one to choose the fastest approximation suitable for a given application.\nIn order of increasing accuracy, they are:\n(maximum error: 0.001)\nwhere\n,\n,\n,\n(maximum error: 0)\nwhere\n,\n,\n,\n(maximum error: 0)\nwhere\n,\n,\n,\n,\n,\n(maximum error: 0)\nwhere\n,\n,\n,\n,\n,\nAll of these approximations are valid for\nx ≥ 0\n. To use these approximations for negative x, use the fact that\nerf x\nis an odd function, so\n. \n \nExponential bounds and a pure exponential approximation for the complementary error function are given by\n \n \nThe above have been generalized to sums of N exponentials with increasing accuracy in terms of N so that\nerfc x\ncan be accurately approximated or bounded by\n2Q̃(√2x)\n, where\nIn particular, there is a systematic methodology to solve the numerical coefficients\n{(an,bn)}N n = 1\nthat yield a [minimax](https://bliptext.com/articles/minimax-approximation-algorithm) approximation or bound for the closely related [Q-function](https://bliptext.com/articles/q-function):\nQ(x) ≈ Q̃(x)\n,\nQ(x) ≤ Q̃(x)\n, or\nQ(x) ≥ Q̃(x)\nfor\nx ≥ 0\n. The coefficients\n{(an,bn)}N n = 1\nfor many variations of the exponential approximations and bounds up to\nhave been released to open access as a comprehensive dataset.\n \n \nA tight approximation of the complementary error function for\nx ∈ [0,∞)\nis given by [Karagiannidis](https://bliptext.com/articles/george-karagiannidis) & Lioumpas (2007) who showed for the appropriate choice of parameters\n{A,B}\nthat\nThey determined\n{A,B} = {1.98,1.135}\n, which gave a good approximation for all\nx ≥ 0\n. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.\n \n \nA single-term lower bound is\nwhere the parameter β can be picked to minimize error on the desired interval of approximation.\n \n \nAnother approximation is given by Sergei Winitzki using his \"global Padé approximations\":\nwhere\nThis is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the relative error is less than 0.00035 for all real x. Using the alternate value\na ≈ 0.147\nreduces the maximum relative error to about 0.00013.\nThis approximation can be inverted to obtain an approximation for the inverse error function:\n \n \nAn approximation with a maximal error of 0 for any real argument is:\nwith\nand\n \n An approximation of with a maximum relative error less than 2^{-53} in absolute value is:\nfor x\\ge 0,\nand for x \n \nA simple approximation for real-valued arguments could be done through [Hyperbolic functions](https://bliptext.com/articles/hyperbolic-functions):\nwhich keeps the absolute difference.\n \n \n\n## Table of values\n\n# Related functions\n\n## Complementary error function\nThe **complementary error function**, denoted\nerfc\n, is defined as\nwhich also defines\nerfcx\n, the **scaled complementary error function** (which can be used instead of\nerfc\nto avoid [arithmetic underflow](https://bliptext.com/articles/arithmetic-underflow) ). Another form of\nerfc x\nfor\nx ≥ 0\nis known as Craig's formula, after its discoverer:\nThis expression is valid only for positive values of x, but it can be used in conjunction with\nto obtain\nerfc(x)\nfor negative values.\nThis form is advantageous in that the range of integration is fixed and finite.\nAn extension of this expression for the\nerfc\nof the sum of two non-negative variables is as follows:\n# [Plot of the complementary error function Erfc(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D | upload.wikimedia.org/wikipedia/commons/4/46/Plot///of///the///complementary///error///function///Erfc%28z%29///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///created///with/ //Mathematica///13.1///function///ComplexPlot3D.svg]\n\n## Imaginary error function\nThe **imaginary error function**, denoted\nerfi\n, is defined as\nwhere\nD(x)\nis the [Dawson function](https://bliptext.com/articles/dawson-function) (which can be used instead of\nerfi\nto avoid [arithmetic overflow](https://bliptext.com/articles/arithmetic-overflow) ).\nDespite the name \"imaginary error function\",\nerfi x\nis real when x is real.\nWhen the error function is evaluated for arbitrary [complex](https://bliptext.com/articles/complex-number) arguments z, the resulting **[complex](https://bliptext.com/articles/complex-number) error function** is usually discussed in scaled form as the [Faddeeva function](https://bliptext.com/articles/faddeeva-function):\n# [Plot of the imaginary error function Erfi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D | upload.wikimedia.org/wikipedia/commons/b/b5/Plot///of///the///imaginary///error///function///Erfi%28z%29///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///created///with///Ma thematica///13.1///function///ComplexPlot3D.svg]\n\n## Cumulative distribution function\nThe error function is essentially identical to the standard [normal cumulative distribution function](https://bliptext.com/articles/normal-cumulative-distribution-function), denoted\nΦ\n, also named\nnorm(x)\nby some software languages, as they differ only by scaling and translation.\nIndeed,\nor rearranged for\nerf\nand\nerfc\nConsequently, the error function is also closely related to the [Q-function](https://bliptext.com/articles/q-function), which is the tail probability of the standard normal distribution.\nThe Q-function can be expressed in terms of the error function as\nThe [inverse](https://bliptext.com/articles/inverse-function) of\nΦ\nis known as the [normal quantile function](https://bliptext.com/articles/quantile-function), or [probit](https://bliptext.com/articles/probit) function and may be expressed in terms of the inverse error function as\nThe standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.\nThe error function is a special case of the [Mittag-Leffler function](https://bliptext.com/articles/mittag-leffler-function), and can also be expressed as a [confluent hypergeometric function](https://bliptext.com/articles/confluent-hypergeometric-function) (Kummer's function):\nIt has a simple expression in terms of the [Fresnel integral](https://bliptext.com/articles/fresnel-integral).\nIn terms of the [regularized gamma function](https://bliptext.com/articles/regularized-gamma-function) P and the [incomplete gamma function](https://bliptext.com/articles/incomplete-gamma-function),\nsgn x\nis the [sign function](https://bliptext.com/articles/sign-function).\n# [the normal cumulative distribution function plotted in the complex plane | upload.wikimedia.org/wikipedia/commons/a/ad/Normal///cumulative///distribution///function///complex///plot///in///Mathematica///13.1///with///ComplexPlot3D.svg]\n\n## Iterated integrals of the complementary error function\nThe iterated integrals of the complementary error function are defined by\nThe general recurrence formula is\nThey have the power series\nfrom which follow the symmetry properties\nand\n\n# Implementations\n\n## As real function of a real argument\n\n## As complex function of a complex argument","revisionCount":0},"hasMore":false,"nextCursor":null}

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