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In mathematics, the `!error function`! (also called the `!Gauss error function`!), often denoted by `!erf`!, is a function e r f : C → → C {\\displaystyle \\mathrm {erf} :\\mathbb {C} \\to \\mathbb {C} } defined as:`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] erf ⁡ ⁡ ( z ) = 2 π π ∫ ∫ 0 z e − − t 2 d t . {\\displaystyle \\operatorname {erf} (z)={\\frac {2}{\\sqrt {\\pi }}}\\int _{0}^{z}e^{-t^{2}}\\,\\mathrm {d} t.}

The integral here is a complex `F33f`_`[contour integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Contour_integration]`_`f which is path-independent because exp ⁡ ⁡ ( − − t 2 ) {\\displaystyle \\exp(-t^{2})} is `F33f`_`[holomorphic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Holomorphic_function]`_`f on the whole complex plane C {\\displaystyle \\mathbb {C} } . In many applications, the function argument is a real number, in which case the function value is also real.

In some old texts,`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f] the error function is defined without the factor of 2 π π {\\displaystyle {\\frac {2}{\\sqrt {\\pi }}}} . This `F33f`_`[nonelementary integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nonelementary_integral]`_`f is a `F33f`_`[sigmoid`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sigmoid_function]`_`f function that occurs often in `F33f`_`[probability`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability]`_`f, `F33f`_`[statistics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Statistics]`_`f, and `F33f`_`[partial differential equations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Partial_differential_equation]`_`f.

In statistics, for non-negative real values of x, the error function has the following interpretation: for a real `F33f`_`[random variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Random_variable]`_`f Y that is `F33f`_`[normally distributed`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_distribution]`_`f with `F33f`_`[mean`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mean]`_`f 0 and `F33f`_`[standard deviation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Standard_deviation]`_`f 1 2 {\\displaystyle {\\frac {1}{\\sqrt {2}}}} , erf(`*x`*) is the probability that Y falls in the range [−`*x`*, `*x`*].

Two closely related functions are the `!complementary error function`! e r f c : C → → C {\\displaystyle \\mathrm {erfc} :\\mathbb {C} \\to \\mathbb {C} } is defined as

erfc ⁡ ⁡ ( z ) = 1 − − erf ⁡ ⁡ ( z ) , {\\displaystyle \\operatorname {erfc} (z)=1-\\operatorname {erf} (z),}

and the `!imaginary error function`! e r f i : C → → C {\\displaystyle \\mathrm {erfi} :\\mathbb {C} \\to \\mathbb {C} } is defined as

erfi ⁡ ⁡ ( z ) = − − i erf ⁡ ⁡ ( i z ) , {\\displaystyle \\operatorname {erfi} (z)=-i\\operatorname {erf} (iz),}

where i is the `F33f`_`[imaginary unit`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Imaginary_unit]`_`f.

>>Contents

• `F0af`_`[Name`#name]`_`f
• `F0af`_`[Applications`#applications]`_`f
• `F0af`_`[Properties`#properties]`_`f
• `F0af`_`[Taylor series`#taylor-series]`_`f
• `F0af`_`[Derivative and integral`#derivative-and-integral]`_`f
• `F0af`_`[Bürmann series`#b-rmann-series]`_`f
• `F0af`_`[Inverse functions`#inverse-functions]`_`f
• `F0af`_`[Asymptotic expansion`#asymptotic-expansion]`_`f
• `F0af`_`[Continued fraction expansion`#continued-fraction-expansion]`_`f
• `F0af`_`[Factorial series`#factorial-series]`_`f
• `F0af`_`[Bounds and Numerical approximations`#bounds-and-numerical-approximations]`_`f
• `F0af`_`[Approximation with elementary functions`#approximation-with-elementary-functions]`_`f
• `F0af`_`[Table of values`#table-of-values]`_`f
• `F0af`_`[Related functions`#related-functions]`_`f
• `F0af`_`[Complementary error function`#complementary-error-function]`_`f
• `F0af`_`[Imaginary error function`#imaginary-error-function]`_`f
• `F0af`_`[Cumulative distribution function`#cumulative-distribution-function]`_`f
• `F0af`_`[Iterated integrals of the complementary error function`#iterated-integrals-of-the-complementary-error-function]`_`f
• `F0af`_`[Implementations`#implementations]`_`f
• `F0af`_`[As real function of a real argument`#as-real-function-of-a-real-argument]`_`f
• `F0af`_`[As complex function of a complex argument`#as-complex-function-of-a-complex-argument]`_`f
• `F0af`_`[References`#references]`_`f
• `F0af`_`[Further reading`#further-reading]`_`f
• `F0af`_`[External links`#external-links]`_`f

-─

>>Name

The name "error function" and its abbreviation erf were proposed by `F33f`_`[J. W. L. Glaisher`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=James_Whitbread_Lee_Glaisher]`_`f in 1871 on account of its connection with "the theory of Probability, and notably the theory of `F33f`_`[Errors`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Errors_and_residuals]`_`f."`:cite-ref-glaisher1871a-3-0[`F5bf`_`[3`#cite-note-glaisher1871a-3]`_`f] The error function complement was also discussed by Glaisher in a separate publication in the same year.`:cite-ref-glaisher1871b-4-0[`F5bf`_`[4`#cite-note-glaisher1871b-4]`_`f] For the "law of facility" of errors whose `F33f`_`[density`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_density]`_`f is given by f ( x ) = ( c π π ) 1 / 2 e − − c x 2 {\\displaystyle f(x)=\\left({\\frac {c}{\\pi }}\\right)^{1/2}e^{-cx^{2}}} (the `F33f`_`[normal distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_distribution]`_`f), Glaisher calculates the probability of an error lying between p and q as: ( c π π ) 1 2 ∫ ∫ p q e − − c x 2 d x = 1 2 ( erf ⁡ ⁡ ( q c ) − − erf ⁡ ⁡ ( p c ) ) . {\\displaystyle \\left({\\frac {c}{\\pi }}\\right)^{\\frac {1}{2}}\\int _{p}^{q}e^{-cx^{2}}\\,\\mathrm {d} x={\\tfrac {1}{2}}\\left(\\operatorname {erf} \\left(q{\\sqrt {c}}\\right)-\\operatorname {erf} \\left(p{\\sqrt {c}}\\right)\\right).}

>>Applications

When the results of a series of measurements are described by a `F33f`_`[normal distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_distribution]`_`f with `F33f`_`[standard deviation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Standard_deviation]`_`f σ and `F33f`_`[expected value`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Expected_value]`_`f 0, then erf (⁠`*a`*/`*σ`* √2⁠) is the probability that the error of a single measurement lies between −`*a`* and +`*a`*, for positive a. This is useful, for example, in determining the `F33f`_`[bit error rate`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bit_error_rate]`_`f of a digital communication system.

The error and complementary error functions occur, for example, in solutions of the `F33f`_`[heat equation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Heat_equation]`_`f when `F33f`_`[boundary conditions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boundary_condition]`_`f are given by the `F33f`_`[Heaviside step function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Heaviside_step_function]`_`f.

The error function and its approximations can be used to estimate results that hold `F33f`_`[with high probability`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=With_high_probability]`_`f or with low probability. Given a random variable `*X`* ~ Norm[`*μ`*,`*σ`*] (a normal distribution with mean μ and standard deviation σ) and a constant `*L`* > `*μ`*, it can be shown via integration by substitution: Pr [ X ≤ ≤ L ] = 1 2 + 1 2 erf ⁡ ⁡ ( L − − μ μ 2 σ σ ) ≈ ≈ A exp ⁡ ⁡ ( − − B ( L − − μ μ σ σ ) 2 ) {\\displaystyle {\\begin{aligned}\\Pr[X\\leq L]&={\\frac {1}{2}}+{\\frac {1}{2}}\\operatorname {erf} \\left({\\frac {L-\\mu }{{\\sqrt {2}}\\sigma }}\\right)\\\\&\\approx A\\exp \\left(-B\\left({\\frac {L-\\mu }{\\sigma }}\\right)^{2}\\right)\\end{aligned}}}

where A and B are certain numeric constants. If L is sufficiently far from the mean, specifically `*μ`* − `*L`* ≥ `*σ`*√ln(`*k`*), then:

Pr [ X ≤ ≤ L ] ≤ ≤ A exp ⁡ ⁡ ( − − B ln ⁡ ⁡ ( k ) ) = A k B {\\displaystyle \\Pr[X\\leq L]\\leq A\\exp(-B\\ln(k))={\\frac {A}{k^{B}}}}

so the probability goes to 0 as `*k`* → ∞.

The probability for X being in the interval [`*La`*, `*Lb`*] can be derived as Pr [ L a ≤ ≤ X ≤ ≤ L b ] = ∫ ∫ L a L b 1 2 π π σ σ exp ⁡ ⁡ ( − − ( x − − μ μ ) 2 2 σ σ 2 ) d x = 1 2 ( erf ⁡ ⁡ ( L b − − μ μ 2 σ σ ) − − erf ⁡ ⁡ ( L a − − μ μ 2 σ σ ) ) . {\\displaystyle {\\begin{aligned}\\Pr[L_{a}\\leq X\\leq L_{b}]&=\\int _{L_{a}}^{L_{b}}{\\frac {1}{{\\sqrt {2\\pi }}\\sigma }}\\exp \\left(-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}\\right)\\,\\mathrm {d} x\\\\&={\\frac {1}{2}}\\left(\\operatorname {erf} \\left({\\frac {L_{b}-\\mu }{{\\sqrt {2}}\\sigma }}\\right)-\\operatorname {erf} \\left({\\frac {L_{a}-\\mu }{{\\sqrt {2}}\\sigma }}\\right)\\right).\\end{aligned}}}

>>Properties

The property erf (−`*z`*) = −erf(`*z`*) means that the error function is an `F33f`_`[odd function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Even_and_odd_functions]`_`f. This directly results from the fact that the integrand `*e`*−`*t`*2 is an `F33f`_`[even function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Even_function]`_`f (the antiderivative of an even function which is zero at the origin is an odd function and vice versa).

Since the error function is an `F33f`_`[entire function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Entire_function]`_`f which takes real numbers to real numbers, for any `F33f`_`[complex number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f z: erf ⁡ ⁡ ( z ¯ ¯ ) = erf ⁡ ⁡ ( z ) ¯ ¯ {\\displaystyle \\operatorname {erf} ({\\overline {z}})={\\overline {\\operatorname {erf} (z)}}} where z ¯ ¯ {\\displaystyle {\\overline {z}}} denotes the `F33f`_`[complex conjugate`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_conjugate]`_`f of z {\\displaystyle z} .

The integrand `*f`* = exp(−`*z`*2) and `*f`* = erf(`*z`*) are shown in the complex z-plane in the figures at right with `F33f`_`[domain coloring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Domain_coloring]`_`f.

The error function at +∞ is exactly 1 (see `F33f`_`[Gaussian integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gaussian_integral]`_`f). At the real axis, erf `*z`* approaches unity at `*z`* → +∞ and −1 at `*z`* → −∞. At the imaginary axis, it tends to ±`*i`*∞.

>>>Taylor series

The error function is an `F33f`_`[entire function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Entire_function]`_`f; it has no singularities (except that at infinity) and its `F33f`_`[Taylor expansion`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Taylor_expansion]`_`f always converges. For `*x`* >> 1, however, cancellation of leading terms makes the Taylor expansion unpractical.

The defining integral cannot be evaluated in `F33f`_`[closed form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Closed-form_expression]`_`f in terms of `F33f`_`[elementary functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elementary_function_(differential_algebra)]`_`f (see `F33f`_`[Liouville's theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Liouville's_theorem_(differential_algebra)]`_`f), but by expanding the `F33f`_`[integrand`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integrand]`_`f `*e`*−`*z`*2 into its `F33f`_`[Maclaurin series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Maclaurin_series]`_`f and integrating term by term, one obtains the error function's Maclaurin series as: erf ⁡ ⁡ ( z ) = 2 π π ∑ ∑ n = 0 ∞ ∞ ( − − 1 ) n z 2 n + 1 n ! ( 2 n + 1 ) = 2 π π ( z − − z 3 3 + z 5 10 − − z 7 42 + z 9 216 − − ⋯ ⋯ ) {\\displaystyle {\\begin{aligned}\\operatorname {erf} (z)&={\\frac {2}{\\sqrt {\\pi }}}\\sum _{n=0}^{\\infty }{\\frac {(-1)^{n}z^{2n+1}}{n!(2n+1)}}\\\\[6pt]&={\\frac {2}{\\sqrt {\\pi }}}\\left(z-{\\frac {z^{3}}{3}}+{\\frac {z^{5}}{10}}-{\\frac {z^{7}}{42}}+{\\frac {z^{9}}{216}}-\\cdots \\right)\\end{aligned}}} which holds for every `F33f`_`[complex number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f z. The denominator terms are sequence A007680 in the `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OEIS]`_`f.

For iterative calculation of the above series, the following alternative formulation may be useful: erf ⁡ ⁡ ( z ) = 2 π π ∑ ∑ n = 0 ∞ ∞ ( z ∏ ∏ k = 1 n − − ( 2 k − − 1 ) z 2 k ( 2 k + 1 ) ) = 2 π π ∑ ∑ n = 0 ∞ ∞ z 2 n + 1 ∏ ∏ k = 1 n − − z 2 k {\\displaystyle {\\begin{aligned}\\operatorname {erf} (z)&={\\frac {2}{\\sqrt {\\pi }}}\\sum _{n=0}^{\\infty }\\left(z\\prod _{k=1}^{n}{\\frac {-(2k-1)z^{2}}{k(2k+1)}}\\right)\\\\[6pt]&={\\frac {2}{\\sqrt {\\pi }}}\\sum _{n=0}^{\\infty }{\\frac {z}{2n+1}}\\prod _{k=1}^{n}{\\frac {-z^{2}}{k}}\\end{aligned}}} because ⁠−(2`*k`* − 1)`*z`*2/`*k`*(2`*k`* + 1)⁠ expresses the multiplier to turn the kth term into the (`*k`* + 1)th term (considering z as the first term).

The imaginary error function has a very similar Maclaurin series, which is: erfi ⁡ ⁡ ( z ) = 2 π π ∑ ∑ n = 0 ∞ ∞ z 2 n + 1 n ! ( 2 n + 1 ) = 2 π π ( z + z 3 3 + z 5 10 + z 7 42 + z 9 216 + ⋯ ⋯ ) {\\displaystyle {\\begin{aligned}\\operatorname {erfi} (z)&={\\frac {2}{\\sqrt {\\pi }}}\\sum _{n=0}^{\\infty }{\\frac {z^{2n+1}}{n!(2n+1)}}\\\\[6pt]&={\\frac {2}{\\sqrt {\\pi }}}\\left(z+{\\frac {z^{3}}{3}}+{\\frac {z^{5}}{10}}+{\\frac {z^{7}}{42}}+{\\frac {z^{9}}{216}}+\\cdots \\right)\\end{aligned}}} which holds for every `F33f`_`[complex number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f z.

>>>Derivative and integral

The derivative of the error function follows immediately from its definition: d d z erf ⁡ ⁡ ( z ) = 2 π π e − − z 2 . {\\displaystyle {\\frac {\\mathrm {d} }{\\mathrm {d} z}}\\operatorname {erf} (z)={\\frac {2}{\\sqrt {\\pi }}}e^{-z^{2}}.} From this, the derivative of the imaginary error function is also immediate: d d z erfi ⁡ ⁡ ( z ) = 2 π π e z 2 . {\\displaystyle {\\frac {d}{dz}}\\operatorname {erfi} (z)={\\frac {2}{\\sqrt {\\pi }}}e^{z^{2}}.} Higher order derivatives are given by erf ( k ) ⁡ ⁡ ( z ) = 2 ( − − 1 ) k − − 1 π π H k − − 1 ( z ) e − − z 2 = 2 π π d k − − 1 d z k − − 1 ( e − − z 2 ) , k = 1 , 2 , … … {\\displaystyle \\operatorname {erf} ^{(k)}(z)={\\frac {2(-1)^{k-1}}{\\sqrt {\\pi }}}{\\mathit {H}}_{k-1}(z)e^{-z^{2}}={\\frac {2}{\\sqrt {\\pi }}}{\\frac {\\mathrm {d} ^{k-1}}{\\mathrm {d} z^{k-1}}}\\left(e^{-z^{2}}\\right),\\qquad k=1,2,\\dots } where H are the physicists' `F33f`_`[Hermite polynomials`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hermite_polynomials]`_`f.`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f]

An `F33f`_`[antiderivative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Antiderivative]`_`f of the error function, obtainable by `F33f`_`[integration by parts`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integration_by_parts]`_`f, is ∫ ∫ erf ⁡ ⁡ ( z ) d z = z erf ⁡ ⁡ ( z ) + e − − z 2 π π + C . {\\displaystyle \\int \\operatorname {erf} (z)dz=z\\operatorname {erf} (z)+{\\frac {e^{-z^{2}}}{\\sqrt {\\pi }}}+C.} An antiderivative of the imaginary error function, also obtainable by integration by parts, is ∫ ∫ erfi ⁡ ⁡ ( z ) d z = z erfi ⁡ ⁡ ( z ) − − e z 2 π π + C . {\\displaystyle \\int \\operatorname {erfi} (z)dz=z\\operatorname {erfi} (z)-{\\frac {e^{z^{2}}}{\\sqrt {\\pi }}}+C.}

>>>Bürmann series

An expansion,`:cite-ref-6[`F5bf`_`[6`#cite-note-6]`_`f] which converges more rapidly for all real values of x than a Taylor expansion, is obtained by using `F33f`_`[Hans Heinrich Bürmann`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hans_Heinrich_Bürmann]`_`f's theorem:`:cite-ref-7[`F5bf`_`[7`#cite-note-7]`_`f] erf ⁡ ⁡ ( x ) = 2 π π sgn ⁡ ⁡ ( x ) ⋅ ⋅ 1 − − e − − x 2 ( 1 − − 1 12 ( 1 − − e − − x 2 ) − − 7 480 ( 1 − − e − − x 2 ) 2 − − 5 896 ( 1 − − e − − x 2 ) 3 − − 787 276480 ( 1 − − e − − x 2 ) 4 − − ⋯ ⋯ ) = 2 π π sgn ⁡ ⁡ ( x ) ⋅ ⋅ 1 − − e − − x 2 ( π π 2 + ∑ ∑ k = 1 ∞ ∞ c k e − − k x 2 ) . {\\displaystyle {\\begin{aligned}\\operatorname {erf} (x)&={\\frac {2}{\\sqrt {\\pi }}}\\operatorname {sgn}(x)\\cdot {\\sqrt {1-e^{-x^{2}}}}\\left(1-{\\frac {1}{12}}\\left(1-e^{-x^{2}}\\right)-{\\frac {7}{480}}\\left(1-e^{-x^{2}}\\right)^{2}-{\\frac {5}{896}}\\left(1-e^{-x^{2}}\\right)^{3}-{\\frac {787}{276480}}\\left(1-e^{-x^{2}}\\right)^{4}-\\cdots \\right)\\\\[10pt]&={\\frac {2}{\\sqrt {\\pi }}}\\operatorname {sgn}(x)\\cdot {\\sqrt {1-e^{-x^{2}}}}\\left({\\frac {\\sqrt {\\pi }}{2}}+\\sum _{k=1}^{\\infty }c_{k}e^{-kx^{2}}\\right).\\end{aligned}}} where sgn is the `F33f`_`[sign function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sign_function]`_`f. By keeping only the first two coefficients and choosing `*c`*1 = ⁠31/200⁠ and `*c`*2 = −⁠341/8000⁠, the resulting approximation shows its largest relative error at `*x`* = ±1.40587, where it is less than 0.0034361: erf ⁡ ⁡ ( x ) ≈ ≈ 2 π π sgn ⁡ ⁡ ( x ) ⋅ ⋅ 1 − − e − − x 2 ( π π 2 + 31 200 e − − x 2 − − 341 8000 e − − 2 x 2 ) . {\\displaystyle \\operatorname {erf} (x)\\approx {\\frac {2}{\\sqrt {\\pi }}}\\operatorname {sgn}(x)\\cdot {\\sqrt {1-e^{-x^{2}}}}\\left({\\frac {\\sqrt {\\pi }}{2}}+{\\frac {31}{200}}e^{-x^{2}}-{\\frac {341}{8000}}e^{-2x^{2}}\\right).}

>>>Inverse functions

Given a complex number z, there is not a `*unique`* complex number w satisfying erf(`*w`*) = `*z`*, so a true inverse function would be multivalued. However, for −1 < `*x`* < 1, there is a unique `*real`* number denoted erf−1(`*x`*) satisfying erf ⁡ ⁡ ( erf − − 1 ⁡ ⁡ ( x ) ) = x . {\\displaystyle \\operatorname {erf} \\left(\\operatorname {erf} ^{-1}(x)\\right)=x.}

The `!inverse error function`! is usually defined with domain (−1,1), and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk |`*z`*| < 1 of the complex plane, using the Maclaurin series`:cite-ref-8[`F5bf`_`[8`#cite-note-8]`_`f] erf − − 1 ⁡ ⁡ ( z ) = ∑ ∑ k = 0 ∞ ∞ c k 2 k + 1 ( π π 2 z ) 2 k + 1 , {\\displaystyle \\operatorname {erf} ^{-1}(z)=\\sum _{k=0}^{\\infty }{\\frac {c_{k}}{2k+1}}\\left({\\frac {\\sqrt {\\pi }}{2}}z\\right)^{2k+1},} where `*c`*0 = 1 and c k = ∑ ∑ m = 0 k − − 1 c m c k − − 1 − − m ( m + 1 ) ( 2 m + 1 ) = { 1 , 1 , 7 6 , 127 90 , 4369 2520 , 34807 16200 , … … } . {\\displaystyle {\\begin{aligned}c_{k}&=\\sum _{m=0}^{k-1}{\\frac {c_{m}c_{k-1-m}}{(m+1)(2m+1)}}\\\\[1ex]&=\\left\\{1,1,{\\frac {7}{6}},{\\frac {127}{90}},{\\frac {4369}{2520}},{\\frac {34807}{16200}},\\ldots \\right\\}.\\end{aligned}}}

So we have the series expansion (common factors have been canceled from numerators and denominators): erf − − 1 ⁡ ⁡ ( z ) = π π 2 ( z + π π 12 z 3 + 7 π π 2 480 z 5 + 127 π π 3 40320 z 7 + 4369 π π 4 5806080 z 9 + 34807 π π 5 182476800 z 11 + ⋯ ⋯ ) . {\\displaystyle \\operatorname {erf} ^{-1}(z)={\\frac {\\sqrt {\\pi }}{2}}\\left(z+{\\frac {\\pi }{12}}z^{3}+{\\frac {7\\pi ^{2}}{480}}z^{5}+{\\frac {127\\pi ^{3}}{40320}}z^{7}+{\\frac {4369\\pi ^{4}}{5806080}}z^{9}+{\\frac {34807\\pi ^{5}}{182476800}}z^{11}+\\cdots \\right).} (After cancellation the numerator and denominator values in `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f: A092676 and `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f: A092677 respectively; without cancellation the numerator terms are values in `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f: A002067.) The error function's value at ±∞ is equal to ±1.

For |`*z`*| < 1, we have erf(erf−1(`*z`*)) = `*z`*.

The `!inverse complementary error function`! is defined as erfc − − 1 ⁡ ⁡ ( 1 − − z ) = erf − − 1 ⁡ ⁡ ( z ) . {\\displaystyle \\operatorname {erfc} ^{-1}(1-z)=\\operatorname {erf} ^{-1}(z).} For real x, there is a unique `*real`* number erfi−1(`*x`*) satisfying erfi(erfi−1(`*x`*)) = `*x`*. The `!inverse imaginary error function`! is defined as erfi−1(`*x`*).`:cite-ref-9[`F5bf`_`[9`#cite-note-9]`_`f]

For any real `*x`*, `F33f`_`[Newton's method`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Newton's_method]`_`f can be used to compute erfi−1(`*x`*), and for −1 ≤ `*x`* ≤ 1, the following Maclaurin series converges: erfi − − 1 ⁡ ⁡ ( z ) = ∑ ∑ k = 0 ∞ ∞ ( − − 1 ) k c k 2 k + 1 ( π π 2 z ) 2 k + 1 , {\\displaystyle \\operatorname {erfi} ^{-1}(z)=\\sum _{k=0}^{\\infty }{\\frac {(-1)^{k}c_{k}}{2k+1}}\\left({\\frac {\\sqrt {\\pi }}{2}}z\\right)^{2k+1},} where `*c`*`*k`* is defined as above.

>>>Asymptotic expansion

A useful `F33f`_`[asymptotic expansion`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Asymptotic_expansion]`_`f of the complementary error function (and therefore also of the error function) for large real x is erfc ⁡ ⁡ ( x ) = e − − x 2 x π π ( 1 + ∑ ∑ n = 1 ∞ ∞ ( − − 1 ) n 1 ⋅ ⋅ 3 ⋅ ⋅ 5 ⋯ ⋯ ( 2 n − − 1 ) ( 2 x 2 ) n ) = e − − x 2 x π π ∑ ∑ n = 0 ∞ ∞ ( − − 1 ) n ( 2 n − − 1 ) ! ! ( 2 x 2 ) n , {\\displaystyle {\\begin{aligned}\\operatorname {erfc} (x)&={\\frac {e^{-x^{2}}}{x{\\sqrt {\\pi }}}}\\left(1+\\sum _{n=1}^{\\infty }(-1)^{n}{\\frac {1\\cdot 3\\cdot 5\\cdots (2n-1)}{\\left(2x^{2}\\right)^{n}}}\\right)\\\\[6pt]&={\\frac {e^{-x^{2}}}{x{\\sqrt {\\pi }}}}\\sum _{n=0}^{\\infty }(-1)^{n}{\\frac {(2n-1)!!}{\\left(2x^{2}\\right)^{n}}},\\end{aligned}}} where (2`*n`* − 1)!! is the `F33f`_`[double factorial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Double_factorial]`_`f of (2`*n`* − 1), which is the product of all odd numbers up to (2`*n`* − 1). This series diverges for every finite x, and its meaning as asymptotic expansion is that for any integer `*N`* ≥ 1 one has erfc ⁡ ⁡ ( x ) = e − − x 2 x π π ∑ ∑ n = 0 N − − 1 ( − − 1 ) n ( 2 n − − 1 ) ! ! ( 2 x 2 ) n + R N ( x ) {\\displaystyle \\operatorname {erfc} (x)={\\frac {e^{-x^{2}}}{x{\\sqrt {\\pi }}}}\\sum _{n=0}^{N-1}(-1)^{n}{\\frac {(2n-1)!!}{\\left(2x^{2}\\right)^{n}}}+R_{N}(x)} where the remainder is R N ( x ) := ( − − 1 ) N ( 2 N − − 1 ) ! ! π π ⋅ ⋅ 2 N − − 1 ∫ ∫ x ∞ ∞ t − − 2 N e − − t 2 d t , {\\displaystyle R_{N}(x):={\\frac {(-1)^{N}\\,(2N-1)!!}{{\\sqrt {\\pi }}\\cdot 2^{N-1}}}\\int _{x}^{\\infty }t^{-2N}e^{-t^{2}}\\,\\mathrm {d} t,} which follows easily by induction, writing e − − t 2 = − − 1 2 t d d t e − − t 2 {\\displaystyle e^{-t^{2}}=-{\\frac {1}{2t}}\\,{\\frac {\\mathrm {d} }{\\mathrm {d} t}}e^{-t^{2}}} and integrating by parts.

The asymptotic behavior of the remainder term, in `F33f`_`[Landau notation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Landau_notation]`_`f, is R N ( x ) = O ( x − − ( 1 + 2 N ) e − − x 2 ) {\\displaystyle R_{N}(x)=O\\left(x^{-(1+2N)}e^{-x^{2}}\\right)} as `*x`* → ∞. This can be found by R N ( x ) ∝ ∝ ∫ ∫ x ∞ ∞ t − − 2 N e − − t 2 d t = e − − x 2 ∫ ∫ 0 ∞ ∞ ( t + x ) − − 2 N e − − t 2 − − 2 t x d t ≤ ≤ e − − x 2 ∫ ∫ 0 ∞ ∞ x − − 2 N e − − 2 t x d t ∝ ∝ x − − ( 1 + 2 N ) e − − x 2 . {\\displaystyle R_{N}(x)\\propto \\int _{x}^{\\infty }t^{-2N}e^{-t^{2}}\\,\\mathrm {d} t=e^{-x^{2}}\\int _{0}^{\\infty }(t+x)^{-2N}e^{-t^{2}-2tx}\\,\\mathrm {d} t\\leq e^{-x^{2}}\\int _{0}^{\\infty }x^{-2N}e^{-2tx}\\,\\mathrm {d} t\\propto x^{-(1+2N)}e^{-x^{2}}.} For large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc `*x`* (while for not too large values of x, the above Taylor expansion at 0 provides a very fast convergence).

>>>Continued fraction expansion

A `F33f`_`[continued fraction`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Continued_fraction]`_`f expansion of the complementary error function was found by `F33f`_`[Laplace`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pierre-Simon_Laplace]`_`f:`:cite-ref-10[`F5bf`_`[10`#cite-note-10]`_`f]`:cite-ref-11[`F5bf`_`[11`#cite-note-11]`_`f] erfc ⁡ ⁡ ( z ) = z π π e − − z 2 1 z 2 + a 1 1 + a 2 z 2 + a 3 1 + ⋯ ⋯ , a m = m 2 . {\\displaystyle \\operatorname {erfc} (z)={\\frac {z}{\\sqrt {\\pi }}}e^{-z^{2}}{\\cfrac {1}{z^{2}+{\\cfrac {a_{1}}{1+{\\cfrac {a_{2}}{z^{2}+{\\cfrac {a_{3}}{1+\\dotsb }}}}}}}},\\qquad a_{m}={\\frac {m}{2}}.}

>>>Factorial series

The inverse factorial series: erfc ⁡ ⁡ ( z ) = e − − z 2 π π z ∑ ∑ n = 0 ∞ ∞ ( − − 1 ) n Q n ( z 2 + 1 ) n ¯ ¯ = e − − z 2 π π z [ 1 − − 1 2 1 ( z 2 + 1 ) + 1 4 1 ( z 2 + 1 ) ( z 2 + 2 ) − − ⋯ ⋯ ] {\\displaystyle {\\begin{aligned}\\operatorname {erfc} (z)&={\\frac {e^{-z^{2}}}{{\\sqrt {\\pi }}\\,z}}\\sum _{n=0}^{\\infty }{\\frac {\\left(-1\\right)^{n}Q_{n}}{{\\left(z^{2}+1\\right)}^{\\bar {n}}}}\\\\[1ex]&={\\frac {e^{-z^{2}}}{{\\sqrt {\\pi }}\\,z}}\\left[1-{\\frac {1}{2}}{\\frac {1}{(z^{2}+1)}}+{\\frac {1}{4}}{\\frac {1}{\\left(z^{2}+1\\right)\\left(z^{2}+2\\right)}}-\\cdots \\right]\\end{aligned}}} converges for Re(`*z`*2) > 0. Here Q n = def 1 Γ Γ ( 1 2 ) ∫ ∫ 0 ∞ ∞ τ τ ( τ τ − − 1 ) ⋯ ⋯ ( τ τ − − n + 1 ) τ τ − − 1 2 e − − τ τ d τ τ = ∑ ∑ k = 0 n ( 1 2 ) k ¯ ¯ s ( n , k ) , {\\displaystyle {\\begin{aligned}Q_{n}&{\\overset {\\text{def}}{{}={}}}{\\frac {1}{\\Gamma {\\left({\\frac {1}{2}}\\right)}}}\\int _{0}^{\\infty }\\tau (\\tau -1)\\cdots (\\tau -n+1)\\tau ^{-{\\frac {1}{2}}}e^{-\\tau }\\,d\\tau \\\\[1ex]&=\\sum _{k=0}^{n}\\left({\\frac {1}{2}}\\right)^{\\bar {k}}s(n,k),\\end{aligned}}} `*z`*`*n`* denotes the `F33f`_`[rising factorial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rising_factorial]`_`f, and `*s`*(`*n`*,`*k`*) denotes a signed `F33f`_`[Stirling number of the first kind`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stirling_number_of_the_first_kind]`_`f.`:cite-ref-12[`F5bf`_`[12`#cite-note-12]`_`f]`:cite-ref-13[`F5bf`_`[13`#cite-note-13]`_`f] There also exists a representation by an infinite sum containing the `F33f`_`[double factorial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Double_factorial]`_`f: erf ⁡ ⁡ ( z ) = 2 π π ∑ ∑ n = 0 ∞ ∞ ( − − 2 ) n ( 2 n − − 1 ) ! ! ( 2 n + 1 ) ! z 2 n + 1 {\\displaystyle \\operatorname {erf} (z)={\\frac {2}{\\sqrt {\\pi }}}\\sum _{n=0}^{\\infty }{\\frac {(-2)^{n}(2n-1)!!}{(2n+1)!}}z^{2n+1}}

>>Bounds and Numerical approximations

>>>Approximation with elementary functions

• `F33f`_`[Abramowitz and Stegun`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Abramowitz_and_Stegun]`_`f give several approximations of varying accuracy (equations 7.1.25–28). This allows one to choose the fastest approximation suitable for a given application. In order of increasing accuracy, they are: erf ⁡ ⁡ ( x ) ≈ ≈ 1 − − 1 ( 1 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 ) 4 , x ≥ ≥ 0 {\\displaystyle \\operatorname {erf} (x)\\approx 1-{\\frac {1}{\\left(1+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}\\right)^{4}}},\\qquad x\\geq 0} (maximum error: 5×10−4) where `*a`*1 = 0.278393, `*a`*2 = 0.230389, `*a`*3 = 0.000972, `*a`*4 = 0.078108 erf ⁡ ⁡ ( x ) ≈ ≈ 1 − − ( a 1 t + a 2 t 2 + a 3 t 3 ) e − − x 2 , t = 1 1 + p x , x ≥ ≥ 0 {\\displaystyle \\operatorname {erf} (x)\\approx 1-\\left(a_{1}t+a_{2}t^{2}+a_{3}t^{3}\\right)e^{-x^{2}},\\quad t={\\frac {1}{1+px}},\\qquad x\\geq 0} (maximum error: 2.5×10−5) where `*p`* = 0.47047, `*a`*1 = 0.3480242, `*a`*2 = −0.0958798, `*a`*3 = 0.7478556 erf ⁡ ⁡ ( x ) ≈ ≈ 1 − − 1 ( 1 + a 1 x + a 2 x 2 + ⋯ ⋯ + a 6 x 6 ) 16 , x ≥ ≥ 0 {\\displaystyle \\operatorname {erf} (x)\\approx 1-{\\frac {1}{\\left(1+a_{1}x+a_{2}x^{2}+\\cdots +a_{6}x^{6}\\right)^{16}}},\\qquad x\\geq 0} (maximum error: 3×10−7) where `*a`*1 = 0.0705230784, `*a`*2 = 0.0422820123, `*a`*3 = 0.0092705272, `*a`*4 = 0.0001520143, `*a`*5 = 0.0002765672, `*a`*6 = 0.0000430638 erf ⁡ ⁡ ( x ) ≈ ≈ 1 − − ( a 1 t + a 2 t 2 + ⋯ ⋯ + a 5 t 5 ) e − − x 2 , t = 1 1 + p x {\\displaystyle \\operatorname {erf} (x)\\approx 1-\\left(a_{1}t+a_{2}t^{2}+\\cdots +a_{5}t^{5}\\right)e^{-x^{2}},\\quad t={\\frac {1}{1+px}}} (maximum error: 1.5×10−7) where `*p`* = 0.3275911, `*a`*1 = 0.254829592, `*a`*2 = −0.284496736, `*a`*3 = 1.421413741, `*a`*4 = −1.453152027, `*a`*5 = 1.061405429 All of these approximations are valid for `*x`* ≥ 0. To use these approximations for negative x, use the fact that erf(`*x`*) is an odd function, so erf(`*x`*) = −erf(−`*x`*).
• Exponential bounds and a pure exponential approximation for the complementary error function are given by`:cite-ref-14[`F5bf`_`[14`#cite-note-14]`_`f] erfc ⁡ ⁡ ( x ) ≤ ≤ 1 2 e − − 2 x 2 + 1 2 e − − x 2 ≤ ≤ e − − x 2 , x > 0 erfc ⁡ ⁡ ( x ) ≈ ≈ 1 6 e − − x 2 + 1 2 e − − 4 3 x 2 , x > 0. {\\displaystyle {\\begin{aligned}\\operatorname {erfc} (x)&\\leq {\\frac {1}{2}}e^{-2x^{2}}+{\\frac {1}{2}}e^{-x^{2}}\\leq e^{-x^{2}},&\\quad x&>0\\\\[1.5ex]\\operatorname {erfc} (x)&\\approx {\\frac {1}{6}}e^{-x^{2}}+{\\frac {1}{2}}e^{-{\\frac {4}{3}}x^{2}},&\\quad x&>0.\\end{aligned}}}
• The above have been generalized to sums of N exponentials`:cite-ref-15[`F5bf`_`[15`#cite-note-15]`_`f] with increasing accuracy in terms of N so that erfc(`*x`*) can be accurately approximated or bounded by 2`*Q̃`*(√2`*x`*), where Q ~ ~ ( x ) = ∑ ∑ n = 1 N a n e − − b n x 2 . {\\displaystyle {\\tilde {Q}}(x)=\\sum _{n=1}^{N}a_{n}e^{-b_{n}x^{2}}.} In particular, there is a systematic methodology to solve the numerical coefficients {(`*an`*,`*bn`*)}`*N`* `*n`* = 1 that yield a `F33f`_`[minimax`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Minimax_approximation_algorithm]`_`f approximation or bound for the closely related `F33f`_`[Q-function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Q-function]`_`f: `*Q`*(`*x`*) ≈ `*Q̃`*(`*x`*), `*Q`*(`*x`*) ≤ `*Q̃`*(`*x`*), or `*Q`*(`*x`*) ≥ `*Q̃`*(`*x`*) for `*x`* ≥ 0. The coefficients {(`*an`*,`*bn`*)}`*N`* `*n`* = 1 for many variations of the exponential approximations and bounds up to `*N`* = 25 have been released to open access as a comprehensive dataset.`:cite-ref-16[`F5bf`_`[16`#cite-note-16]`_`f]
• A tight approximation of the complementary error function for `*x`* ∈ [0,∞) is given by `F33f`_`[Karagiannidis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=George_Karagiannidis]`_`f & Lioumpas (2007)`:cite-ref-17[`F5bf`_`[17`#cite-note-17]`_`f] who showed for the appropriate choice of parameters {`*A`*,`*B`*} that erfc ⁡ ⁡ ( x ) ≈ ≈ ( 1 − − e − − A x ) e − − x 2 B π π x . {\\displaystyle \\operatorname {erfc} (x)\\approx {\\frac {\\left(1-e^{-Ax}\\right)e^{-x^{2}}}{B{\\sqrt {\\pi }}x}}.} They determined {`*A`*,`*B`*} = {1.98,1.135}, which gave a good approximation for all `*x`* ≥ 0. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.`:cite-ref-18[`F5bf`_`[18`#cite-note-18]`_`f]
• A single-term lower bound is`:cite-ref-19[`F5bf`_`[19`#cite-note-19]`_`f] erfc ⁡ ⁡ ( x ) ≥ ≥ 2 e π π β β − − 1 β β e − − β β x 2 , x ≥ ≥ 0 , β β > 1 , {\\displaystyle \\operatorname {erfc} (x)\\geq {\\sqrt {\\frac {2e}{\\pi }}}{\\frac {\\sqrt {\\beta -1}}{\\beta }}e^{-\\beta x^{2}},\\qquad x\\geq 0,\\quad \\beta >1,} where the parameter β can be picked to minimize error on the desired interval of approximation.
• Another approximation is given by Sergei Winitzki using his "global Padé approximations":`:cite-ref-20[`F5bf`_`[20`#cite-note-20]`_`f]`:cite-ref-21[`F5bf`_`[21`#cite-note-21]`_`f] erf ⁡ ⁡ ( x ) ≈ ≈ sgn ⁡ ⁡ x ⋅ ⋅ 1 − − exp ⁡ ⁡ ( − − x 2 4 π π + a x 2 1 + a x 2 ) {\\displaystyle \\operatorname {erf} (x)\\approx \\operatorname {sgn} x\\cdot {\\sqrt {1-\\exp \\left(-x^{2}{\\frac {{\\frac {4}{\\pi }}+ax^{2}}{1+ax^{2}}}\\right)}}} where a = 8 ( π π − − 3 ) 3 π π ( 4 − − π π ) ≈ ≈ 0.140012. {\\displaystyle a={\\frac {8(\\pi -3)}{3\\pi (4-\\pi )}}\\approx 0.140012.} This 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 `*a`* ≈ 0.147 reduces the maximum relative error to about 0.00013.`:cite-ref-22[`F5bf`_`[22`#cite-note-22]`_`f] This approximation can be inverted to obtain an approximation for the inverse error function: erf − − 1 ⁡ ⁡ ( x ) ≈ ≈ sgn ⁡ ⁡ x ⋅ ⋅ ( 2 π π a + ln ⁡ ⁡ ( 1 − − x 2 ) 2 ) 2 − − ln ⁡ ⁡ ( 1 − − x 2 ) a − − ( 2 π π a + ln ⁡ ⁡ ( 1 − − x 2 ) 2 ) . {\\displaystyle \\operatorname {erf} ^{-1}(x)\\approx \\operatorname {sgn} x\\cdot {\\sqrt {{\\sqrt {\\left({\\frac {2}{\\pi a}}+{\\frac {\\ln \\left(1-x^{2}\\right)}{2}}\\right)^{2}-{\\frac {\\ln \\left(1-x^{2}\\right)}{a}}}}-\\left({\\frac {2}{\\pi a}}+{\\frac {\\ln \\left(1-x^{2}\\right)}{2}}\\right)}}.}
• An approximation with a maximal error of 1.2×10−7 for any real argument is:`:cite-ref-23[`F5bf`_`[23`#cite-note-23]`_`f] erf ⁡ ⁡ ( x ) = { 1 − − τ τ x ≥ ≥ 0 τ τ − − 1 x < 0 {\\displaystyle \\operatorname {erf} (x)={\\begin{cases}1-\\tau &x\\geq 0\\\\\\tau -1&x<0\\end{cases}}} with τ τ = t ⋅ ⋅ exp ⁡ ⁡ ( − − x 2 − − 1.26551223 + 1.00002368 t + 0.37409196 t 2 + 0.09678418 t 3 − − 0.18628806 t 4 + 0.27886807 t 5 − − 1.13520398 t 6 + 1.48851587 t 7 − − 0.82215223 t 8 + 0.17087277 t 9 ) {\\displaystyle {\\begin{aligned}\\tau &=t\\cdot \\exp \\left(-x^{2}-1.26551223+1.00002368t+0.37409196t^{2}+0.09678418t^{3}-0.18628806t^{4}\\right.\\\\&\\left.\\qquad \\qquad \\qquad +0.27886807t^{5}-1.13520398t^{6}+1.48851587t^{7}-0.82215223t^{8}+0.17087277t^{9}\\right)\\end{aligned}}} and t = 1 1 + 1 2 | x | . {\\displaystyle t={\\frac {1}{1+{\\frac {1}{2}}|x|}}.}
• An approximation of erfc {\\displaystyle \\operatorname {erfc} } with a maximum relative error less than 2 − − 53 {\\displaystyle 2^{-53}} ( ≈ ≈ 1.1 × × 10 − − 16 ) {\\displaystyle \\left(\\approx 1.1\\times 10^{-16}\\right)} in absolute value is:`:cite-ref-24[`F5bf`_`[24`#cite-note-24]`_`f] for x ≥ ≥ 0 {\\displaystyle x\\geq 0} , erfc ⁡ ⁡ ( x ) = ( 0.56418958354775629 x + 2.06955023132914151 ) ( x 2 + 2.71078540045147805 x + 5.80755613130301624 x 2 + 3.47954057099518960 x + 12.06166887286239555 ) ( x 2 + 3.47469513777439592 x + 12.07402036406381411 x 2 + 3.72068443960225092 x + 8.44319781003968454 ) ( x 2 + 4.00561509202259545 x + 9.30596659485887898 x 2 + 3.90225704029924078 x + 6.36161630953880464 ) ( x 2 + 5.16722705817812584 x + 9.12661617673673262 x 2 + 4.03296893109262491 x + 5.13578530585681539 ) ( x 2 + 5.95908795446633271 x + 9.19435612886969243 x 2 + 4.11240942957450885 x + 4.48640329523408675 ) e − − x 2 {\\displaystyle {\\begin{aligned}\\operatorname {erfc} \\left(x\\right)&=\\left({\\frac {0.56418958354775629}{x+2.06955023132914151}}\\right)\\left({\\frac {x^{2}+2.71078540045147805x+5.80755613130301624}{x^{2}+3.47954057099518960x+12.06166887286239555}}\\right)\\\\&\\left({\\frac {x^{2}+3.47469513777439592x+12.07402036406381411}{x^{2}+3.72068443960225092x+8.44319781003968454}}\\right)\\left({\\frac {x^{2}+4.00561509202259545x+9.30596659485887898}{x^{2}+3.90225704029924078x+6.36161630953880464}}\\right)\\\\&\\left({\\frac {x^{2}+5.16722705817812584x+9.12661617673673262}{x^{2}+4.03296893109262491x+5.13578530585681539}}\\right)\\left({\\frac {x^{2}+5.95908795446633271x+9.19435612886969243}{x^{2}+4.11240942957450885x+4.48640329523408675}}\\right)e^{-x^{2}}\\\\\\end{aligned}}} and for x < 0 {\\displaystyle x<0} erfc ⁡ ⁡ ( x ) = 2 − − erfc ⁡ ⁡ ( − − x ) {\\displaystyle \\operatorname {erfc} \\left(x\\right)=2-\\operatorname {erfc} \\left(-x\\right)}
• A simple approximation for real-valued arguments could be done through `F33f`_`[Hyperbolic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hyperbolic_functions]`_`f: erf ⁡ ⁡ ( x ) ≈ ≈ z ( x ) = tanh ⁡ ⁡ ( 2 π π ( x + 11 123 x 3 ) ) {\\displaystyle \\operatorname {erf} \\left(x\\right)\\approx z(x)=\\tanh \\left({\\frac {2}{\\sqrt {\\pi }}}\\left(x+{\\frac {11}{123}}x^{3}\\right)\\right)} which keeps the absolute difference | erf ⁡ ⁡ ( x ) − − z ( x ) | < 0.000358 , ∀ ∀ x {\\displaystyle \\left|\\operatorname {erf} \\left(x\\right)-z(x)\\right|<0.000358,\\,\\forall x} .
• Since the error function and the Gaussian Q-function are closely related through the identity erfc ⁡ ⁡ ( x ) = 2 Q ( 2 x ) {\\displaystyle \\operatorname {erfc} (x)=2Q({\\sqrt {2}}x)} or equivalently Q ( x ) = 1 2 erfc ⁡ ⁡ ( x 2 ) {\\displaystyle Q(x)={\\frac {1}{2}}\\operatorname {erfc} \\left({\\frac {x}{\\sqrt {2}}}\\right)} , bounds developed for the Q-function can be adapted to approximate the complementary error function. A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments x ∈ ∈ [ 0 , ∞ ∞ ) {\\displaystyle x\\in [0,\\infty )} was introduced by Abreu (2012)`:cite-ref-25[`F5bf`_`[25`#cite-note-25]`_`f] based on a simple algebraic expression with only two exponential terms: Q ( x ) ≥ ≥ 1 12 e − − x 2 + 1 2 π π ( x + 1 ) e − − x 2 / 2 , x ≥ ≥ 0 , {\\displaystyle Q(x)\\geq {\\frac {1}{12}}e^{-x^{2}}+{\\frac {1}{{\\sqrt {2\\pi }}(x+1)}}e^{-x^{2}/2},\\qquad x\\geq 0,} and Q ( x ) ≤ ≤ 1 50 e − − x 2 + 1 2 ( x + 1 ) e − − x 2 / 2 , x ≥ ≥ 0. {\\displaystyle Q(x)\\leq {\\frac {1}{50}}e^{-x^{2}}+{\\frac {1}{2(x+1)}}e^{-x^{2}/2},\\qquad x\\geq 0.} These bounds stem from a unified form Q B ( x ; a , b ) = exp ⁡ ⁡ ( − − x 2 ) a + exp ⁡ ⁡ ( − − x 2 / 2 ) b ( x + 1 ) , {\\displaystyle Q_{\\mathrm {B} }(x;a,b)={\\frac {\\exp(-x^{2})}{a}}+{\\frac {\\exp(-x^{2}/2)}{b(x+1)}},} where the parameters a {\\displaystyle a} and b {\\displaystyle b} are selected to ensure the bounding properties: for the lower bound, a L = 12 {\\displaystyle a_{\\mathrm {L} }=12} and b L = 2 π π {\\displaystyle b_{\\mathrm {L} }={\\sqrt {2\\pi }}} , and for the upper bound, a U = 50 {\\displaystyle a_{\\mathrm {U} }=50} and b U = 2 {\\displaystyle b_{\\mathrm {U} }=2} . These expressions maintain simplicity and tightness, providing a practical trade-off between accuracy and ease of computation. They are particularly valuable in theoretical contexts, such as communication theory over fading channels, where both functions frequently appear. Additionally, the original Q-function bounds can be extended to Q n ( x ) {\\displaystyle Q^{n}(x)} for positive integers n {\\displaystyle n} via the binomial theorem, suggesting potential adaptability for powers of erfc ⁡ ⁡ ( x ) {\\displaystyle \\operatorname {erfc} (x)} , though this is less commonly required in error function applications.

>>>Table of values

`t
| x | erf( x ) | 1 − erf( x ) |
|---|---|---|
| 0 | 0 | 1 |
| 0.02 | 0.022 564 575 | 0.977 435 425 |
| 0.04 | 0.045 111 106 | 0.954 888 894 |
| 0.06 | 0.067 621 594 | 0.932 378 406 |
| 0.08 | 0.090 078 126 | 0.909 921 874 |
| 0.1 | 0.112 462 916 | 0.887 537 084 |
| 0.2 | 0.222 702 589 | 0.777 297 411 |
| 0.3 | 0.328 626 759 | 0.671 373 241 |
| 0.4 | 0.428 392 355 | 0.571 607 645 |
| 0.5 | 0.520 499 878 | 0.479 500 122 |
| 0.6 | 0.603 856 091 | 0.396 143 909 |
| 0.7 | 0.677 801 194 | 0.322 198 806 |
| 0.8 | 0.742 100 965 | 0.257 899 035 |
| 0.9 | 0.796 908 212 | 0.203 091 788 |
| 1 | 0.842 700 793 | 0.157 299 207 |
| 1.1 | 0.880 205 070 | 0.119 794 930 |
| 1.2 | 0.910 313 978 | 0.089 686 022 |
| 1.3 | 0.934 007 945 | 0.065 992 055 |
| 1.4 | 0.952 285 120 | 0.047 714 880 |
| 1.5 | 0.966 105 146 | 0.033 894 854 |
| 1.6 | 0.976 348 383 | 0.023 651 617 |
| 1.7 | 0.983 790 459 | 0.016 209 541 |
| 1.8 | 0.989 090 502 | 0.010 909 498 |
| 1.9 | 0.992 790 429 | 0.007 209 571 |
| 2 | 0.995 322 265 | 0.004 677 735 |
| 2.1 | 0.997 020 533 | 0.002 979 467 |
| 2.2 | 0.998 137 154 | 0.001 862 846 |
| 2.3 | 0.998 856 823 | 0.001 143 177 |
| 2.4 | 0.999 311 486 | 0.000 688 514 |
| 2.5 | 0.999 593 048 | 0.000 406 952 |
| 3 | 0.999 977 910 | 0.000 022 090 |
| 3.5 | 0.999 999 257 | 0.000 000 743 |
`t

>>Related functions

>>>Complementary error function

The `!complementary error function`!, denoted erfc, is defined as

erfc ⁡ ⁡ ( x ) = 1 − − erf ⁡ ⁡ ( x ) = 2 π π ∫ ∫ x ∞ ∞ e − − t 2 d t = e − − x 2 erfcx ⁡ ⁡ ( x ) , {\\displaystyle {\\begin{aligned}\\operatorname {erfc} (x)&=1-\\operatorname {erf} (x)\\\\[5pt]&={\\frac {2}{\\sqrt {\\pi }}}\\int _{x}^{\\infty }e^{-t^{2}}\\,\\mathrm {d} t\\\\[5pt]&=e^{-x^{2}}\\operatorname {erfcx} (x),\\end{aligned}}} which also defines erfcx, the `!scaled complementary error function`!`:cite-ref-cody93-26-0[`F5bf`_`[26`#cite-note-cody93-26]`_`f] (which can be used instead of erfc to avoid `F33f`_`[arithmetic underflow`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arithmetic_underflow]`_`f`:cite-ref-cody93-26-1[`F5bf`_`[26`#cite-note-cody93-26]`_`f]`:cite-ref-zaghloul07-27-0[`F5bf`_`[27`#cite-note-zaghloul07-27]`_`f]). Another form of erfc `*x`* for `*x`* ≥ 0 is known as Craig's formula, after its discoverer:`:cite-ref-28[`F5bf`_`[28`#cite-note-28]`_`f] erfc ⁡ ⁡ ( x ∣ ∣ x ≥ ≥ 0 ) = 2 π π ∫ ∫ 0 π π 2 exp ⁡ ⁡ ( − − x 2 sin 2 ⁡ ⁡ θ θ ) d θ θ . {\\displaystyle \\operatorname {erfc} (x\\mid x\\geq 0)={\\frac {2}{\\pi }}\\int _{0}^{\\frac {\\pi }{2}}\\exp \\left(-{\\frac {x^{2}}{\\sin ^{2}\\theta }}\\right)\\,\\mathrm {d} \\theta .} This expression is valid only for positive values of x, but it can be used in conjunction with erfc(`*x`*) = 2 − erfc(−`*x`*) to obtain erfc(`*x`*) for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for the erfc of the sum of two non-negative variables is as follows:`:cite-ref-29[`F5bf`_`[29`#cite-note-29]`_`f] erfc ⁡ ⁡ ( x + y ∣ ∣ x , y ≥ ≥ 0 ) = 2 π π ∫ ∫ 0 π π 2 exp ⁡ ⁡ ( − − x 2 sin 2 ⁡ ⁡ θ θ − − y 2 cos 2 ⁡ ⁡ θ θ ) d θ θ . {\\displaystyle \\operatorname {erfc} (x+y\\mid x,y\\geq 0)={\\frac {2}{\\pi }}\\int _{0}^{\\frac {\\pi }{2}}\\exp \\left(-{\\frac {x^{2}}{\\sin ^{2}\\theta }}-{\\frac {y^{2}}{\\cos ^{2}\\theta }}\\right)\\,\\mathrm {d} \\theta .}

>>>Imaginary error function

The `!imaginary error function`!, denoted erfi, is defined as

erfi ⁡ ⁡ ( x ) = − − i erf ⁡ ⁡ ( i x ) = 2 π π ∫ ∫ 0 x e t 2 d t = 2 π π e x 2 D ( x ) , {\\displaystyle {\\begin{aligned}\\operatorname {erfi} (x)&=-i\\operatorname {erf} (ix)\\\\[5pt]&={\\frac {2}{\\sqrt {\\pi }}}\\int _{0}^{x}e^{t^{2}}\\,\\mathrm {d} t\\\\[5pt]&={\\frac {2}{\\sqrt {\\pi }}}e^{x^{2}}D(x),\\end{aligned}}} where `*D`*(`*x`*) is the `F33f`_`[Dawson function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dawson_function]`_`f (which can be used instead of erfi to avoid `F33f`_`[arithmetic overflow`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arithmetic_overflow]`_`f`:cite-ref-cody93-26-2[`F5bf`_`[26`#cite-note-cody93-26]`_`f]).

Despite the name "imaginary error function", erfi(`*x`*) is real when x is real.

When the error function is evaluated for arbitrary `F33f`_`[complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f arguments z, the resulting `!complex error function`! is usually discussed in scaled form as the `F33f`_`[Faddeeva function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Faddeeva_function]`_`f: w ( z ) = e − − z 2 erfc ⁡ ⁡ ( − − i z ) = erfcx ⁡ ⁡ ( − − i z ) . {\\displaystyle w(z)=e^{-z^{2}}\\operatorname {erfc} (-iz)=\\operatorname {erfcx} (-iz).}

>>>Cumulative distribution function

The error function is essentially identical to the standard `F33f`_`[normal cumulative distribution function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_cumulative_distribution_function]`_`f, denoted Φ, also named norm(`*x`*) by some software languages, as they differ only by scaling and translation. Indeed,

Φ Φ ( x ) = 1 2 π π ∫ ∫ − − ∞ ∞ x e − − t 2 2 d t = 1 2 ( 1 + erf ⁡ ⁡ ( x 2 ) ) = 1 2 erfc ⁡ ⁡ ( − − x 2 ) {\\displaystyle {\\begin{aligned}\\Phi (x)&={\\frac {1}{\\sqrt {2\\pi }}}\\int _{-\\infty }^{x}e^{\\tfrac {-t^{2}}{2}}\\,\\mathrm {d} t\\\\[6pt]&={\\frac {1}{2}}\\left(1+\\operatorname {erf} \\left({\\frac {x}{\\sqrt {2}}}\\right)\\right)\\\\[6pt]&={\\frac {1}{2}}\\operatorname {erfc} \\left(-{\\frac {x}{\\sqrt {2}}}\\right)\\end{aligned}}} or rearranged for erf and erfc: erf ⁡ ⁡ ( x ) = 2 Φ Φ ( x 2 ) − − 1 erfc ⁡ ⁡ ( x ) = 2 Φ Φ ( − − x 2 ) = 2 ( 1 − − Φ Φ ( x 2 ) ) . {\\displaystyle {\\begin{aligned}\\operatorname {erf} (x)&=2\\Phi {\\left(x{\\sqrt {2}}\\right)}-1\\\\[6pt]\\operatorname {erfc} (x)&=2\\Phi {\\left(-x{\\sqrt {2}}\\right)}\\\\&=2\\left(1-\\Phi {\\left(x{\\sqrt {2}}\\right)}\\right).\\end{aligned}}}

Consequently, the error function is also closely related to the `F33f`_`[Q-function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Q-function]`_`f, which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as Q ( x ) = 1 2 − − 1 2 erf ⁡ ⁡ ( x 2 ) = 1 2 erfc ⁡ ⁡ ( x 2 ) . {\\displaystyle {\\begin{aligned}Q(x)&={\\frac {1}{2}}-{\\frac {1}{2}}\\operatorname {erf} \\left({\\frac {x}{\\sqrt {2}}}\\right)\\\\&={\\frac {1}{2}}\\operatorname {erfc} \\left({\\frac {x}{\\sqrt {2}}}\\right).\\end{aligned}}}

The `F33f`_`[inverse`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inverse_function]`_`f of Φ is known as the `F33f`_`[normal quantile function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quantile_function]`_`f, or `F33f`_`[probit`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probit]`_`f function and may be expressed in terms of the inverse error function as probit ⁡ ⁡ ( p ) = Φ Φ − − 1 ( p ) = 2 erf − − 1 ⁡ ⁡ ( 2 p − − 1 ) = − − 2 erfc − − 1 ⁡ ⁡ ( 2 p ) . {\\displaystyle \\operatorname {probit} (p)=\\Phi ^{-1}(p)={\\sqrt {2}}\\operatorname {erf} ^{-1}(2p-1)=-{\\sqrt {2}}\\operatorname {erfc} ^{-1}(2p).}

The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.

The error function is a special case of the `F33f`_`[Mittag-Leffler function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mittag-Leffler_function]`_`f, and can also be expressed as a `F33f`_`[confluent hypergeometric function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Confluent_hypergeometric_function]`_`f (Kummer's function): erf ⁡ ⁡ ( x ) = 2 x π π M ( 1 2 , 3 2 , − − x 2 ) . {\\displaystyle \\operatorname {erf} (x)={\\frac {2x}{\\sqrt {\\pi }}}M\\left({\\tfrac {1}{2}},{\\tfrac {3}{2}},-x^{2}\\right).}

It has a simple expression in terms of the `F33f`_`[Fresnel integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fresnel_integral]`_`f.

In terms of the `F33f`_`[regularized gamma function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Regularized_gamma_function]`_`f P and the `F33f`_`[incomplete gamma function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Incomplete_gamma_function]`_`f, erf ⁡ ⁡ ( x ) = sgn ⁡ ⁡ ( x ) ⋅ ⋅ P ( 1 2 , x 2 ) = sgn ⁡ ⁡ ( x ) π π γ γ ( 1 2 , x 2 ) . {\\displaystyle \\operatorname {erf} (x)=\\operatorname {sgn}(x)\\cdot P\\left({\\tfrac {1}{2}},x^{2}\\right)={\\frac {\\operatorname {sgn}(x)}{\\sqrt {\\pi }}}\\gamma {\\left({\\tfrac {1}{2}},x^{2}\\right)}.} sgn(`*x`*) is the `F33f`_`[sign function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sign_function]`_`f.

>>>Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by`:cite-ref-30[`F5bf`_`[30`#cite-note-30]`_`f] i n erfc ⁡ ⁡ ( z ) = ∫ ∫ z ∞ ∞ i n − − 1 erfc ⁡ ⁡ ( ζ ζ ) d ζ ζ i 0 erfc ⁡ ⁡ ( z ) = erfc ⁡ ⁡ ( z ) i 1 erfc ⁡ ⁡ ( z ) = ierfc ⁡ ⁡ ( z ) = 1 π π e − − z 2 − − z erfc ⁡ ⁡ ( z ) i 2 erfc ⁡ ⁡ ( z ) = 1 4 ( erfc ⁡ ⁡ ( z ) − − 2 z ierfc ⁡ ⁡ ( z ) ) {\\displaystyle {\\begin{aligned}i^{n}\\!\\operatorname {erfc} (z)&=\\int _{z}^{\\infty }i^{n-1}\\!\\operatorname {erfc} (\\zeta )\\,\\mathrm {d} \\zeta \\\\[6pt]i^{0}\\!\\operatorname {erfc} (z)&=\\operatorname {erfc} (z)\\\\i^{1}\\!\\operatorname {erfc} (z)&=\\operatorname {ierfc} (z)={\\frac {1}{\\sqrt {\\pi }}}e^{-z^{2}}-z\\operatorname {erfc} (z)\\\\i^{2}\\!\\operatorname {erfc} (z)&={\\tfrac {1}{4}}\\left(\\operatorname {erfc} (z)-2z\\operatorname {ierfc} (z)\\right)\\\\\\end{aligned}}}

The general recurrence formula is 2 n ⋅ ⋅ i n erfc ⁡ ⁡ ( z ) = i n − − 2 erfc ⁡ ⁡ ( z ) − − 2 z ⋅ ⋅ i n − − 1 erfc ⁡ ⁡ ( z ) {\\displaystyle 2n\\cdot i^{n}\\!\\operatorname {erfc} (z)=i^{n-2}\\!\\operatorname {erfc} (z)-2z\\cdot i^{n-1}\\!\\operatorname {erfc} (z)}

They have the power series i n erfc ⁡ ⁡ ( z ) = ∑ ∑ j = 0 ∞ ∞ ( − − z ) j 2 n − − j j ! Γ Γ ( 1 + n − − j 2 ) , {\\displaystyle i^{n}\\!\\operatorname {erfc} (z)=\\sum _{j=0}^{\\infty }{\\frac {(-z)^{j}}{2^{n-j}j!\\,\\Gamma \\left(1+{\\frac {n-j}{2}}\\right)}},} from which follow the symmetry properties i 2 m erfc ⁡ ⁡ ( − − z ) = − − i 2 m erfc ⁡ ⁡ ( z ) + ∑ ∑ q = 0 m z 2 q 2 2 ( m − − q ) − − 1 ( 2 q ) ! ( m − − q ) ! {\\displaystyle i^{2m}\\!\\operatorname {erfc} (-z)=-i^{2m}\\!\\operatorname {erfc} (z)+\\sum _{q=0}^{m}{\\frac {z^{2q}}{2^{2(m-q)-1}(2q)!(m-q)!}}} and i 2 m + 1 erfc ⁡ ⁡ ( − − z ) = i 2 m + 1 erfc ⁡ ⁡ ( z ) + ∑ ∑ q = 0 m z 2 q + 1 2 2 ( m − − q ) − − 1 ( 2 q + 1 ) ! ( m − − q ) ! . {\\displaystyle i^{2m+1}\\!\\operatorname {erfc} (-z)=i^{2m+1}\\!\\operatorname {erfc} (z)+\\sum _{q=0}^{m}{\\frac {z^{2q+1}}{2^{2(m-q)-1}(2q+1)!(m-q)!}}.}

>>Implementations

>>>As real function of a real argument

• In `F33f`_`[POSIX`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=POSIX]`_`f-compliant operating systems, the header `B100`F9d9`F33f`_`[math.h`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Math.h]`_`f`f`b shall declare and the mathematical library `B100`F9d9`F33f`_`[libm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Libm]`_`f`f`b shall provide the functions `B100`F9d9erf`f`b and `B100`F9d9erfc`f`b (`F33f`_`[double precision`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Double_precision]`_`f) as well as their `F33f`_`[single precision`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Single_precision]`_`f and `F33f`_`[extended precision`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Extended_precision]`_`f counterparts `B100`F9d9erff`f`b, `B100`F9d9erfl`f`b and `B100`F9d9erfcf`f`b, `B100`F9d9erfcl`f`b.`:cite-ref-31[`F5bf`_`[31`#cite-note-31]`_`f]
• The `F33f`_`[GNU Scientific Library`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=GNU_Scientific_Library]`_`f provides `B100`F9d9erf`f`b, `B100`F9d9erfc`f`b, `B100`F9d9log(erf)`f`b, and scaled error functions.`:cite-ref-32[`F5bf`_`[32`#cite-note-32]`_`f]

>>>As complex function of a complex argument

• `B100`F9d9libcerf`f`b, numeric C library for complex error functions, provides the complex functions `B100`F9d9cerf`f`b, `B100`F9d9cerfc`f`b, `B100`F9d9cerfcx`f`b and the real functions `B100`F9d9erfi`f`b, `B100`F9d9erfcx`f`b with approximately 13–14 digits precision, based on the `F33f`_`[Faddeeva function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Faddeeva_function]`_`f as implemented in the MIT Faddeeva Package

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citerefandrews1998`aAndrews, Larry C. (1998). `*Special functions of mathematics for engineers`*. SPIE Press. p. 110. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9780819426161.
`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `:citerefwhittakerwatson2021`a`F33f`_`[Whittaker, Edmund Taylor`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Edmund_T._Whittaker]`_`f; `F33f`_`[Watson, George Neville`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=George_N._Watson]`_`f (2021). `F33f`_`[Moll, Victor Hugo`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Victor_Hugo_Moll]`_`f (ed.). `F33f`_`[A Course of Modern Analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=A_Course_of_Modern_Analysis]`_`f (5th revised ed.). `F33f`_`[Cambridge University Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cambridge_University_Press]`_`f. p. 358. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-316-51893-9.
`:cite-note-glaisher1871a-3`!3.`! `F0af`_`[↑`#cite-ref-glaisher1871a-3-0]`_`f `:citerefglaisher1871`aGlaisher, James Whitbread Lee (July 1871). "On a class of definite integrals". `*London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science`*. 4. `!42`! (277): 294–302. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1080/14786447108640568. Retrieved 6 December 2017.
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>>Further reading

• `:citerefabramowitzstegun1983`a`F33f`_`[Abramowitz, Milton`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Milton_Abramowitz]`_`f; `F33f`_`[Stegun, Irene Ann`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Irene_Stegun]`_`f, eds. (1983) [June 1964]. "Chapter 7". `F33f`_`[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Abramowitz_and_Stegun]`_`f. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 297. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-486-61272-0. `F33f`_`[LCCN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=LCCN_(identifier)]`_`f 64-60036. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0167642. `F33f`_`[LCCN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=LCCN_(identifier)]`_`f 65-12253.
• `:citerefpressteukolskyvetterlingflannery2007`aPress, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), "Section 6.2. Incomplete Gamma Function and Error Function", `*Numerical Recipes: The Art of Scientific Computing`* (3rd ed.), New York: Cambridge University Press, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-521-88068-8, archived from the original on 11 August 2011, retrieved 9 August 2011
• `:citereftemme2010`aTemme, Nico M. (2010), "Error Functions, Dawson's and Fresnel Integrals", in `F33f`_`[Olver, Frank W. J.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Frank_W._J._Olver]`_`f; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), `*`F33f`_`[NIST Handbook of Mathematical Functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Digital_Library_of_Mathematical_Functions]`_`f`*, Cambridge University Press, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-521-19225-5, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 2723248.

>>External links

• A Table of Integrals of the Error Functions

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