Интеграл ошибок erf

Error function
Plot of the error function
General information
General definition	$\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t$
Fields of application	Probability, thermodynamics
Domain, Codomain and Image
Domain	$\mathbb {R}$
Image	$\left(-1,1\right)$
Basic features
Parity	Odd
Specific features
Root	0
Derivative	${\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}$
Antiderivative	$\int \operatorname {erf} z\,dz=z\operatorname {erf} z+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}+C$
Series definition
Taylor series	$\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z}{2n+1}}\prod _{k=1}^{n}{\frac {-z^{2}}{k}}$

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as:^[1]

$\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t.$

Some authors define $\operatorname {erf}$ without the factor of $2/{\sqrt {\pi }}$ .^[2]
This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.

In statistics, for non-negative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard deviation 1/√2, erf x is the probability that Y falls in the range [−x, x].

Two closely related functions are the complementary error function (erfc) defined as

$\operatorname {erfc} z=1-\operatorname {erf} z,$

and the imaginary error function (erfi) defined as

$\operatorname {erfi} z=-i\operatorname {erf} iz,$

where i is the imaginary unit.

Name[edit]

The name «error function» and its abbreviation erf were proposed by J. W. L. Glaisher in 1871 on account of its connection with «the theory of Probability, and notably the theory of Errors.»^[3] The error function complement was also discussed by Glaisher in a separate publication in the same year.^[4]
For the «law of facility» of errors whose density is given by

$f(x)=\left({\frac {c}{\pi }}\right)^{\frac {1}{2}}e^{-cx^{2}}$

(the normal distribution), Glaisher calculates the probability of an error lying between p and q as:

$\left({\frac {c}{\pi }}\right)^{\frac {1}{2}}\int _{p}^{q}e^{-cx^{2}}\,\mathrm {d} x={\tfrac {1}{2}}\left(\operatorname {erf} \left(q{\sqrt {c}}\right)-\operatorname {erf} \left(p{\sqrt {c}}\right)\right).$

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

Applications[edit]

When the results of a series of measurements are described by a normal distribution with standard deviation σ and expected value 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 bit error rate of a digital communication system.

The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

The error function and its approximations can be used to estimate results that hold with high probability 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:

${\begin{aligned}\Pr[X\leq L]&={\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} {\frac {L-\mu }{{\sqrt {2}}\sigma }}\\&\approx A\exp \left(-B\left({\frac {L-\mu }{\sigma }}\right)^{2}\right)\end{aligned}}$

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

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

so the probability goes to 0 as k → ∞.

The probability for X being in the interval [L_a, L_b] can be derived as

${\begin{aligned}\Pr[L_{a}\leq X\leq L_{b}]&=\int _{L_{a}}^{L_{b}}{\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\,\mathrm {d} x\\&={\frac {1}{2}}\left(\operatorname {erf} {\frac {L_{b}-\mu }{{\sqrt {2}}\sigma }}-\operatorname {erf} {\frac {L_{a}-\mu }{{\sqrt {2}}\sigma }}\right).\end{aligned}}$

Properties[edit]

Integrand exp(−z²)

erf z

The property erf (−z) = −erf z means that the error function is an odd function. This directly results from the fact that the integrand e^−t² is an even function (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 entire function which takes real numbers to real numbers, for any complex number z:

$\operatorname {erf} {\overline {z}}={\overline {\operatorname {erf} z}}$

where z is the complex conjugate of z.

The integrand f = exp(−z²) and f = erf z are shown in the complex z-plane in the figures at right with domain coloring.

The error function at +∞ is exactly 1 (see Gaussian integral). At the real axis, erf z approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

Taylor series[edit]

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges, but is famously known «[…] for its bad convergence if x > 1.»^[5]

The defining integral cannot be evaluated in closed form in terms of elementary functions (see Liouville’s theorem), but by expanding the integrand e^−z² into its Maclaurin series and integrating term by term, one obtains the error function’s Maclaurin series as:

${\begin{aligned}\operatorname {erf} z&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{n!(2n+1)}}\\[6pt]&={\frac {2}{\sqrt {\pi }}}\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{10}}-{\frac {z^{7}}{42}}+{\frac {z^{9}}{216}}-\cdots \right)\end{aligned}}$

which holds for every complex number z. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful:

${\begin{aligned}\operatorname {erf} z&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }\left(z\prod _{k=1}^{n}{\frac {-(2k-1)z^{2}}{k(2k+1)}}\right)\\[6pt]&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z}{2n+1}}\prod _{k=1}^{n}{\frac {-z^{2}}{k}}\end{aligned}}$

because −(2k − 1)z²/k(2k + 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:

${\begin{aligned}\operatorname {erfi} z&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z^{2n+1}}{n!(2n+1)}}\\[6pt]&={\frac {2}{\sqrt {\pi }}}\left(z+{\frac {z^{3}}{3}}+{\frac {z^{5}}{10}}+{\frac {z^{7}}{42}}+{\frac {z^{9}}{216}}+\cdots \right)\end{aligned}}$

which holds for every complex number z.

Derivative and integral[edit]

The derivative of the error function follows immediately from its definition:

${\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}.$

From this, the derivative of the imaginary error function is also immediate:

${\frac {d}{dz}}\operatorname {erfi} z={\frac {2}{\sqrt {\pi }}}e^{z^{2}}.$

An antiderivative of the error function, obtainable by integration by parts, is

$z\operatorname {erf} z+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}.$

An antiderivative of the imaginary error function, also obtainable by integration by parts, is

$z\operatorname {erfi} z-{\frac {e^{z^{2}}}{\sqrt {\pi }}}.$

Higher order derivatives are given by

$\operatorname {erf} ^{(k)}z={\frac {2(-1)^{k-1}}{\sqrt {\pi }}}{\mathit {H}}_{k-1}(z)e^{-z^{2}}={\frac {2}{\sqrt {\pi }}}{\frac {\mathrm {d} ^{k-1}}{\mathrm {d} z^{k-1}}}\left(e^{-z^{2}}\right),\qquad k=1,2,\dots$

where H are the physicists’ Hermite polynomials.^[6]

Bürmann series[edit]

An expansion,^[7] which converges more rapidly for all real values of x than a Taylor expansion, is obtained by using Hans Heinrich Bürmann’s theorem:^[8]

${\begin{aligned}\operatorname {erf} x&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn} x\cdot {\sqrt {1-e^{-x^{2}}}}\left(1-{\frac {1}{12}}\left(1-e^{-x^{2}}\right)-{\frac {7}{480}}\left(1-e^{-x^{2}}\right)^{2}-{\frac {5}{896}}\left(1-e^{-x^{2}}\right)^{3}-{\frac {787}{276480}}\left(1-e^{-x^{2}}\right)^{4}-\cdots \right)\\[10pt]&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn} x\cdot {\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+\sum _{k=1}^{\infty }c_{k}e^{-kx^{2}}\right).\end{aligned}}$

where sgn is the sign function. By keeping only the first two coefficients and choosing c₁ = 31/200 and c₂ = −341/8000, the resulting approximation shows its largest relative error at x = ±1.3796, where it is less than 0.0036127:

$\operatorname {erf} x\approx {\frac {2}{\sqrt {\pi }}}\operatorname {sgn} x\cdot {\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+{\frac {31}{200}}e^{-x^{2}}-{\frac {341}{8000}}e^{-2x^{2}}\right).$

Inverse functions[edit]

Inverse error function

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

$\operatorname {erf} \left(\operatorname {erf} ^{-1}x\right)=x.$

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^[9]

$\operatorname {erf} ^{-1}z=\sum _{k=0}^{\infty }{\frac {c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},$

where c₀ = 1 and

${\begin{aligned}c_{k}&=\sum _{m=0}^{k-1}{\frac {c_{m}c_{k-1-m}}{(m+1)(2m+1)}}\\&=\left\{1,1,{\frac {7}{6}},{\frac {127}{90}},{\frac {4369}{2520}},{\frac {34807}{16200}},\ldots \right\}.\end{aligned}}$

So we have the series expansion (common factors have been canceled from numerators and denominators):

$\operatorname {erf} ^{-1}z={\frac {\sqrt {\pi }}{2}}\left(z+{\frac {\pi }{12}}z^{3}+{\frac {7\pi ^{2}}{480}}z^{5}+{\frac {127\pi ^{3}}{40320}}z^{7}+{\frac {4369\pi ^{4}}{5806080}}z^{9}+{\frac {34807\pi ^{5}}{182476800}}z^{11}+\cdots \right).$

(After cancellation the numerator/denominator fractions are entries OEIS: A092676/OEIS: A092677 in the OEIS; without cancellation the numerator terms are given in entry OEIS: 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

$\operatorname {erfc} ^{-1}(1-z)=\operatorname {erf} ^{-1}z.$

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.^[10]

For any real x, Newton’s method can be used to compute erfi⁻¹ x, and for −1 ≤ x ≤ 1, the following Maclaurin series converges:

$\operatorname {erfi} ^{-1}z=\sum _{k=0}^{\infty }{\frac {(-1)^{k}c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},$

where c_k is defined as above.

Asymptotic expansion[edit]

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real x is

${\begin{aligned}\operatorname {erfc} x&={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\left(1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {1\cdot 3\cdot 5\cdots (2n-1)}{\left(2x^{2}\right)^{n}}}\right)\\[6pt]&={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n-1)!!}{\left(2x^{2}\right)^{n}}},\end{aligned}}$

where (2n − 1)!! is the double factorial of (2n − 1), which is the product of all odd numbers up to (2n − 1). This series diverges for every finite x, and its meaning as asymptotic expansion is that for any integer N ≥ 1 one has

$\operatorname {erfc} x={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\sum _{n=0}^{N-1}(-1)^{n}{\frac {(2n-1)!!}{\left(2x^{2}\right)^{n}}}+R_{N}(x)$

where the remainder is

$R_{N}(x):={\frac {(-1)^{N}}{\sqrt {\pi }}}2^{1-2N}{\frac {(2N)!}{N!}}\int _{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t,$

which follows easily by induction, writing

$e^{-t^{2}}=-(2t)^{-1}\left(e^{-t^{2}}\right)'$

and integrating by parts.

The asymptotic behavior of the remainder term, in Landau notation, is

$R_{N}(x)=O\left(x^{-(1+2N)}e^{-x^{2}}\right)$

as x → ∞. This can be found by

$R_{N}(x)\propto \int _{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t=e^{-x^{2}}\int _{0}^{\infty }(t+x)^{-2N}e^{-t^{2}-2tx}\,\mathrm {d} t\leq e^{-x^{2}}\int _{0}^{\infty }x^{-2N}e^{-2tx}\,\mathrm {d} t\propto x^{-(1+2N)}e^{-x^{2}}.$

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[edit]

A continued fraction expansion of the complementary error function is:^[11]

$\operatorname {erfc} z={\frac {z}{\sqrt {\pi }}}e^{-z^{2}}{\cfrac {1}{z^{2}+{\cfrac {a_{1}}{1+{\cfrac {a_{2}}{z^{2}+{\cfrac {a_{3}}{1+\dotsb }}}}}}}},\qquad a_{m}={\frac {m}{2}}.$

Integral of error function with Gaussian density function[edit]

$\int _{-\infty }^{\infty }\operatorname {erf} \left(ax+b\right){\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\,\mathrm {d} x=\operatorname {erf} {\frac {a\mu +b}{\sqrt {1+2a^{2}\sigma ^{2}}}},\qquad a,b,\mu ,\sigma \in \mathbb {R}$

which appears related to Ng and Geller, formula 13 in section 4.3^[12] with a change of variables.

Factorial series[edit]

The inverse factorial series:

${\begin{aligned}\operatorname {erfc} z&={\frac {e^{-z^{2}}}{{\sqrt {\pi }}\,z}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}Q_{n}}{{(z^{2}+1)}^{\bar {n}}}}\\&={\frac {e^{-z^{2}}}{{\sqrt {\pi }}\,z}}\left(1-{\frac {1}{2}}{\frac {1}{(z^{2}+1)}}+{\frac {1}{4}}{\frac {1}{(z^{2}+1)(z^{2}+2)}}-\cdots \right)\end{aligned}}$

converges for Re(z²) > 0. Here

${\begin{aligned}Q_{n}&{\overset {\text{def}}{{}={}}}{\frac {1}{\Gamma \left({\frac {1}{2}}\right)}}\int _{0}^{\infty }\tau (\tau -1)\cdots (\tau -n+1)\tau ^{-{\frac {1}{2}}}e^{-\tau }\,d\tau \\&=\sum _{k=0}^{n}\left({\tfrac {1}{2}}\right)^{\bar {k}}s(n,k),\end{aligned}}$

zⁿ denotes the rising factorial, and s(n,k) denotes a signed Stirling number of the first kind.^[13]^[14]
There also exists a representation by an infinite sum containing the double factorial:

$\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-2)^{n}(2n-1)!!}{(2n+1)!}}z^{2n+1}$

Numerical approximations[edit]

Approximation with elementary functions[edit]

Abramowitz and Stegun 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:

$\operatorname {erf} x\approx 1-{\frac {1}{\left(1+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}\right)^{4}}},\qquad x\geq 0$

(maximum error: 5×10⁻⁴)

where a₁ = 0.278393, a₂ = 0.230389, a₃ = 0.000972, a₄ = 0.078108

$\operatorname {erf} x\approx 1-\left(a_{1}t+a_{2}t^{2}+a_{3}t^{3}\right)e^{-x^{2}},\quad t={\frac {1}{1+px}},\qquad x\geq 0$

(maximum error: 2.5×10⁻⁵)

where p = 0.47047, a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556

$\operatorname {erf} x\approx 1-{\frac {1}{\left(1+a_{1}x+a_{2}x^{2}+\cdots +a_{6}x^{6}\right)^{16}}},\qquad x\geq 0$

(maximum error: 3×10⁻⁷)

where a₁ = 0.0705230784, a₂ = 0.0422820123, a₃ = 0.0092705272, a₄ = 0.0001520143, a₅ = 0.0002765672, a₆ = 0.0000430638

$\operatorname {erf} x\approx 1-\left(a_{1}t+a_{2}t^{2}+\cdots +a_{5}t^{5}\right)e^{-x^{2}},\quad t={\frac {1}{1+px}}$

(maximum error: 1.5×10⁻⁷)

where p = 0.3275911, a₁ = 0.254829592, a₂ = −0.284496736, a₃ = 1.421413741, a₄ = −1.453152027, a₅ = 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^[15]

${\begin{aligned}\operatorname {erfc} x&\leq {\tfrac {1}{2}}e^{-2x^{2}}+{\tfrac {1}{2}}e^{-x^{2}}\leq e^{-x^{2}},&\quad x&>0\\\operatorname {erfc} x&\approx {\tfrac {1}{6}}e^{-x^{2}}+{\tfrac {1}{2}}e^{-{\frac {4}{3}}x^{2}},&\quad x&>0.\end{aligned}}$
The above have been generalized to sums of N exponentials^[16] with increasing accuracy in terms of N so that erfc x can be accurately approximated or bounded by 2Q̃(√2x), where

${\tilde {Q}}(x)=\sum _{n=1}^{N}a_{n}e^{-b_{n}x^{2}}.$

In particular, there is a systematic methodology to solve the numerical coefficients {(a_n,b_n)}^N
_{n = 1} that yield a minimax approximation or bound for the closely related Q-function: Q(x) ≈ Q̃(x), Q(x) ≤ Q̃(x), or Q(x) ≥ Q̃(x) for x ≥ 0. The coefficients {(a_n,b_n)}^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.^[17]
A tight approximation of the complementary error function for x ∈ [0,∞) is given by Karagiannidis & Lioumpas (2007)^[18] who showed for the appropriate choice of parameters {A,B} that

$\operatorname {erfc} x\approx {\frac {\left(1-e^{-Ax}\right)e^{-x^{2}}}{B{\sqrt {\pi }}x}}.$

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.^[19]
A single-term lower bound is^[20]

$\operatorname {erfc} x\geq {\sqrt {\frac {2e}{\pi }}}{\frac {\sqrt {\beta -1}}{\beta }}e^{-\beta x^{2}},\qquad x\geq 0,\quad \beta >1,$

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»:^[21]^[22]^: 2–3

$\operatorname {erf} x\approx \operatorname {sgn} x\cdot {\sqrt {1-\exp \left(-x^{2}{\frac {{\frac {4}{\pi }}+ax^{2}}{1+ax^{2}}}\right)}}$

where

$a={\frac {8(\pi -3)}{3\pi (4-\pi )}}\approx 0.140012.$

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.^[23]

This approximation can be inverted to obtain an approximation for the inverse error function:

$\operatorname {erf} ^{-1}x\approx \operatorname {sgn} x\cdot {\sqrt {{\sqrt {\left({\frac {2}{\pi a}}+{\frac {\ln \left(1-x^{2}\right)}{2}}\right)^{2}-{\frac {\ln \left(1-x^{2}\right)}{a}}}}-\left({\frac {2}{\pi a}}+{\frac {\ln \left(1-x^{2}\right)}{2}}\right)}}.$
An approximation with a maximal error of 1.2×10⁻⁷ for any real argument is:^[24]

$\operatorname {erf} x={\begin{cases}1-\tau &x\geq 0\\\tau -1&x<0\end{cases}}$

with

${\begin{aligned}\tau &=t\cdot \exp \left(-x^{2}-1.26551223+1.00002368t+0.37409196t^{2}+0.09678418t^{3}-0.18628806t^{4}\right.\\&\left.\qquad \qquad \qquad +0.27886807t^{5}-1.13520398t^{6}+1.48851587t^{7}-0.82215223t^{8}+0.17087277t^{9}\right)\end{aligned}}$

and

$t={\frac {1}{1+{\frac {1}{2}}|x|}}.$
An approximation of $\operatorname {erfc}$ with a maximum relative error less than $2^{-53}$ $\left(\approx 1.1\times 10^{-16}\right)$ in absolute value is:^[25]
for $x\geq 0$ ,

${\begin{aligned}\operatorname {erfc} \left(x\right)&=\left({\frac {0.56418958354775629}{x+2.06955023132914151}}\right)\left({\frac {x^{2}+2.71078540045147805x+5.80755613130301624}{x^{2}+3.47954057099518960x+12.06166887286239555}}\right)\\&\left({\frac {x^{2}+3.47469513777439592x+12.07402036406381411}{x^{2}+3.72068443960225092x+8.44319781003968454}}\right)\left({\frac {x^{2}+4.00561509202259545x+9.30596659485887898}{x^{2}+3.90225704029924078x+6.36161630953880464}}\right)\\&\left({\frac {x^{2}+5.16722705817812584x+9.12661617673673262}{x^{2}+4.03296893109262491x+5.13578530585681539}}\right)\left({\frac {x^{2}+5.95908795446633271x+9.19435612886969243}{x^{2}+4.11240942957450885x+4.48640329523408675}}\right)e^{-x^{2}}\\\end{aligned}}$

and for

$\operatorname {erfc} \left(x\right)=2-\operatorname {erfc} \left(-x\right)$

Table of values[edit]

x	erf x	1 − erf x
0	0	1
0.02	0.022564575	0.977435425
0.04	0.045111106	0.954888894
0.06	0.067621594	0.932378406
0.08	0.090078126	0.909921874
0.1	0.112462916	0.887537084
0.2	0.222702589	0.777297411
0.3	0.328626759	0.671373241
0.4	0.428392355	0.571607645
0.5	0.520499878	0.479500122
0.6	0.603856091	0.396143909
0.7	0.677801194	0.322198806
0.8	0.742100965	0.257899035
0.9	0.796908212	0.203091788
1	0.842700793	0.157299207
1.1	0.880205070	0.119794930
1.2	0.910313978	0.089686022
1.3	0.934007945	0.065992055
1.4	0.952285120	0.047714880
1.5	0.966105146	0.033894854
1.6	0.976348383	0.023651617
1.7	0.983790459	0.016209541
1.8	0.989090502	0.010909498
1.9	0.992790429	0.007209571
2	0.995322265	0.004677735
2.1	0.997020533	0.002979467
2.2	0.998137154	0.001862846
2.3	0.998856823	0.001143177
2.4	0.999311486	0.000688514
2.5	0.999593048	0.000406952
3	0.999977910	0.000022090
3.5	0.999999257	0.000000743

[edit]

Complementary error function[edit]

The complementary error function, denoted erfc, is defined as

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

${\begin{aligned}\operatorname {erfc} x&=1-\operatorname {erf} x\\[5pt]&={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-t^{2}}\,\mathrm {d} t\\[5pt]&=e^{-x^{2}}\operatorname {erfcx} x,\end{aligned}}$

which also defines erfcx, the scaled complementary error function^[26] (which can be used instead of erfc to avoid arithmetic underflow^[26]^[27]). Another form of erfc x for x ≥ 0 is known as Craig’s formula, after its discoverer:^[28]

$\operatorname {erfc} (x\mid x\geq 0)={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{\sin ^{2}\theta }}\right)\,\mathrm {d} \theta .$

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:^[29]

$\operatorname {erfc} (x+y\mid x,y\geq 0)={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{\sin ^{2}\theta }}-{\frac {y^{2}}{\cos ^{2}\theta }}\right)\,\mathrm {d} \theta .$

Imaginary error function[edit]

The imaginary error function, denoted erfi, is defined as

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

${\begin{aligned}\operatorname {erfi} x&=-i\operatorname {erf} ix\\[5pt]&={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{t^{2}}\,\mathrm {d} t\\[5pt]&={\frac {2}{\sqrt {\pi }}}e^{x^{2}}D(x),\end{aligned}}$

where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow^[26]).

Despite the name «imaginary error function», erfi x is real when x is real.

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

$w(z)=e^{-z^{2}}\operatorname {erfc} (-iz)=\operatorname {erfcx} (-iz).$

Cumulative distribution function[edit]

The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x) by some software languages^{[citation needed]}, as they differ only by scaling and translation. Indeed,

the normal cumulative distribution function plotted in the complex plane

${\begin{aligned}\Phi (x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{\tfrac {-t^{2}}{2}}\,\mathrm {d} t\\[6pt]&={\frac {1}{2}}\left(1+\operatorname {erf} {\frac {x}{\sqrt {2}}}\right)\\[6pt]&={\frac {1}{2}}\operatorname {erfc} \left(-{\frac {x}{\sqrt {2}}}\right)\end{aligned}}$

or rearranged for erf and erfc:

${\begin{aligned}\operatorname {erf} (x)&=2\Phi \left(x{\sqrt {2}}\right)-1\\[6pt]\operatorname {erfc} (x)&=2\Phi \left(-x{\sqrt {2}}\right)\\&=2\left(1-\Phi \left(x{\sqrt {2}}\right)\right).\end{aligned}}$

Consequently, the error function is also closely related to the Q-function, which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as

${\begin{aligned}Q(x)&={\frac {1}{2}}-{\frac {1}{2}}\operatorname {erf} {\frac {x}{\sqrt {2}}}\\&={\frac {1}{2}}\operatorname {erfc} {\frac {x}{\sqrt {2}}}.\end{aligned}}$

The inverse of Φ is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as

$\operatorname {probit} (p)=\Phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)=-{\sqrt {2}}\operatorname {erfc} ^{-1}(2p).$

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 Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer’s function):

$\operatorname {erf} x={\frac {2x}{\sqrt {\pi }}}M\left({\tfrac {1}{2}},{\tfrac {3}{2}},-x^{2}\right).$

It has a simple expression in terms of the Fresnel integral.^{[further explanation needed]}

In terms of the regularized gamma function P and the incomplete gamma function,

$\operatorname {erf} x=\operatorname {sgn} x\cdot P\left({\tfrac {1}{2}},x^{2}\right)={\frac {\operatorname {sgn} x}{\sqrt {\pi }}}\gamma \left({\tfrac {1}{2}},x^{2}\right).$

sgn x is the sign function.

Generalized error functions[edit]

Graph of generalised error functions E_n(x):
grey curve: E₁(x) = 1 − e^−x/√π
red curve: E₂(x) = erf(x)
green curve: E₃(x)
blue curve: E₄(x)
gold curve: E₅(x).

Some authors discuss the more general functions:^{[citation needed]}

$E_{n}(x)={\frac {n!}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{n}}\,\mathrm {d} t={\frac {n!}{\sqrt {\pi }}}\sum _{p=0}^{\infty }(-1)^{p}{\frac {x^{np+1}}{(np+1)p!}}.$

Notable cases are:

E₀(x) is a straight line through the origin: E₀(x) = x/e√π
E₂(x) is the error function, erf x.

After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n > 0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x > 0 using the gamma function and incomplete gamma function:

$E_{n}(x)={\frac {1}{\sqrt {\pi }}}\Gamma (n)\left(\Gamma \left({\frac {1}{n}}\right)-\Gamma \left({\frac {1}{n}},x^{n}\right)\right),\qquad x>0.$

Therefore, we can define the error function in terms of the incomplete gamma function:

$\operatorname {erf} x=1-{\frac {1}{\sqrt {\pi }}}\Gamma \left({\tfrac {1}{2}},x^{2}\right).$

Iterated integrals of the complementary error function[edit]

The iterated integrals of the complementary error function are defined by^[30]

${\begin{aligned}\operatorname {i} ^{n}\!\operatorname {erfc} z&=\int _{z}^{\infty }\operatorname {i} ^{n-1}\!\operatorname {erfc} \zeta \,\mathrm {d} \zeta \\[6pt]\operatorname {i} ^{0}\!\operatorname {erfc} z&=\operatorname {erfc} z\\\operatorname {i} ^{1}\!\operatorname {erfc} z&=\operatorname {ierfc} z={\frac {1}{\sqrt {\pi }}}e^{-z^{2}}-z\operatorname {erfc} z\\\operatorname {i} ^{2}\!\operatorname {erfc} z&={\tfrac {1}{4}}\left(\operatorname {erfc} z-2z\operatorname {ierfc} z\right)\\\end{aligned}}$

The general recurrence formula is

$2n\cdot \operatorname {i} ^{n}\!\operatorname {erfc} z=\operatorname {i} ^{n-2}\!\operatorname {erfc} z-2z\cdot \operatorname {i} ^{n-1}\!\operatorname {erfc} z$

They have the power series

$\operatorname {i} ^{n}\!\operatorname {erfc} z=\sum _{j=0}^{\infty }{\frac {(-z)^{j}}{2^{n-j}j!\,\Gamma \left(1+{\frac {n-j}{2}}\right)}},$

from which follow the symmetry properties

$\operatorname {i} ^{2m}\!\operatorname {erfc} (-z)=-\operatorname {i} ^{2m}\!\operatorname {erfc} z+\sum _{q=0}^{m}{\frac {z^{2q}}{2^{2(m-q)-1}(2q)!(m-q)!}}$

and

$\operatorname {i} ^{2m+1}\!\operatorname {erfc} (-z)=\operatorname {i} ^{2m+1}\!\operatorname {erfc} z+\sum _{q=0}^{m}{\frac {z^{2q+1}}{2^{2(m-q)-1}(2q+1)!(m-q)!}}.$

Implementations[edit]

As real function of a real argument[edit]

In POSIX-compliant operating systems, the header math.h shall declare and the mathematical library libm shall provide the functions erf and erfc (double precision) as well as their single precision and extended precision counterparts erff, erfl and erfcf, erfcl.^[31]

The GNU Scientific Library provides erf, erfc, log(erf), and scaled error functions.^[32]

As complex function of a complex argument[edit]

libcerf, numeric C library for complex error functions, provides the complex functions cerf, cerfc, cerfcx and the real functions erfi, erfcx with approximately 13–14 digits precision, based on the Faddeeva function as implemented in the MIT Faddeeva Package

References[edit]

^ Andrews, Larry C. (1998). Special functions of mathematics for engineers. SPIE Press. p. 110. ISBN 9780819426161.
^ Whittaker, E. T.; Watson, G. N. (1927). A Course of Modern Analysis. Cambridge University Press. p. 341. ISBN 978-0-521-58807-2.
^ Glaisher, 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. doi:10.1080/14786447108640568. Retrieved 6 December 2017.
^ Glaisher, James Whitbread Lee (September 1871). «On a class of definite integrals. Part II». London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 4. 42 (279): 421–436. doi:10.1080/14786447108640600. Retrieved 6 December 2017.
^ «A007680 – OEIS». oeis.org. Retrieved 2 April 2020.
^ Weisstein, Eric W. «Erf». MathWorld.
^ Schöpf, H. M.; Supancic, P. H. (2014). «On Bürmann’s Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion». The Mathematica Journal. 16. doi:10.3888/tmj.16-11.
^ Weisstein, Eric W. «Bürmann’s Theorem». MathWorld.
^ Dominici, Diego (2006). «Asymptotic analysis of the derivatives of the inverse error function». arXiv:math/0607230.
^ Bergsma, Wicher (2006). «On a new correlation coefficient, its orthogonal decomposition and associated tests of independence». arXiv:math/0604627.
^ Cuyt, Annie A. M.; Petersen, Vigdis B.; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). Handbook of Continued Fractions for Special Functions. Springer-Verlag. ISBN 978-1-4020-6948-2.
^ Ng, Edward W.; Geller, Murray (January 1969). «A table of integrals of the Error functions». Journal of Research of the National Bureau of Standards Section B. 73B (1): 1. doi:10.6028/jres.073B.001.
^ Schlömilch, Oskar Xavier (1859). «Ueber facultätenreihen». Zeitschrift für Mathematik und Physik (in German). 4: 390–415.
^ Nielson, Niels (1906). Handbuch der Theorie der Gammafunktion (in German). Leipzig: B. G. Teubner. p. 283 Eq. 3. Retrieved 4 December 2017.
^ Chiani, M.; Dardari, D.; Simon, M.K. (2003). «New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels» (PDF). IEEE Transactions on Wireless Communications. 2 (4): 840–845. CiteSeerX 10.1.1.190.6761. doi:10.1109/TWC.2003.814350.
^ Tanash, I.M.; Riihonen, T. (2020). «Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials». IEEE Transactions on Communications. 68 (10): 6514–6524. arXiv:2007.06939. doi:10.1109/TCOMM.2020.3006902. S2CID 220514754.
^ Tanash, I.M.; Riihonen, T. (2020). «Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]». Zenodo. doi:10.5281/zenodo.4112978.
^ Karagiannidis, G. K.; Lioumpas, A. S. (2007). «An improved approximation for the Gaussian Q-function» (PDF). IEEE Communications Letters. 11 (8): 644–646. doi:10.1109/LCOMM.2007.070470. S2CID 4043576.
^ Tanash, I.M.; Riihonen, T. (2021). «Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function». IEEE Communications Letters. 25 (5): 1468–1471. arXiv:2101.07631. doi:10.1109/LCOMM.2021.3052257. S2CID 231639206.
^ Chang, Seok-Ho; Cosman, Pamela C.; Milstein, Laurence B. (November 2011). «Chernoff-Type Bounds for the Gaussian Error Function». IEEE Transactions on Communications. 59 (11): 2939–2944. doi:10.1109/TCOMM.2011.072011.100049. S2CID 13636638.
^ Winitzki, Sergei (2003). «Uniform approximations for transcendental functions». Computational Science and Its Applications – ICCSA 2003. Lecture Notes in Computer Science. Vol. 2667. Springer, Berlin. pp. 780–789. doi:10.1007/3-540-44839-X_82. ISBN 978-3-540-40155-1.
^ Zeng, Caibin; Chen, Yang Cuan (2015). «Global Padé approximations of the generalized Mittag-Leffler function and its inverse». Fractional Calculus and Applied Analysis. 18 (6): 1492–1506. arXiv:1310.5592. doi:10.1515/fca-2015-0086. S2CID 118148950. Indeed, Winitzki [32] provided the so-called global Padé approximation
^ Winitzki, Sergei (6 February 2008). «A handy approximation for the error function and its inverse» (Document).
^ Numerical Recipes in Fortran 77: The Art of Scientific Computing (ISBN 0-521-43064-X), 1992, page 214, Cambridge University Press.
^ Dia, Yaya D. (2023). Approximate Incomplete Integrals, Application to Complementary Error Function. Available at SSRN: https://ssrn.com/abstract=4487559 or http://dx.doi.org/10.2139/ssrn.4487559, 2023
^ ^a ^b ^c Cody, W. J. (March 1993), «Algorithm 715: SPECFUN—A portable FORTRAN package of special function routines and test drivers» (PDF), ACM Trans. Math. Softw., 19 (1): 22–32, CiteSeerX 10.1.1.643.4394, doi:10.1145/151271.151273, S2CID 5621105
^ Zaghloul, M. R. (1 March 2007), «On the calculation of the Voigt line profile: a single proper integral with a damped sine integrand», Monthly Notices of the Royal Astronomical Society, 375 (3): 1043–1048, Bibcode:2007MNRAS.375.1043Z, doi:10.1111/j.1365-2966.2006.11377.x
^ John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations Archived 3 April 2012 at the Wayback Machine, Proceedings of the 1991 IEEE Military Communication Conference, vol. 2, pp. 571–575.
^ Behnad, Aydin (2020). «A Novel Extension to Craig’s Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis». IEEE Transactions on Communications. 68 (7): 4117–4125. doi:10.1109/TCOMM.2020.2986209. S2CID 216500014.
^ Carslaw, H. S.; Jaeger, J. C. (1959), Conduction of Heat in Solids (2nd ed.), Oxford University Press, ISBN 978-0-19-853368-9, p 484
^ «math.h — mathematical declarations». opengroup.org. 2018. Retrieved 21 April 2023.
^ «Special Functions – GSL 2.7 documentation».

External links[edit]

A Table of Integrals of the Error Functions

Источник

Функция ошибок

Аргумент функции ошибок erf(x)

Функция ошибок

Дополнительная функция ошибок

Функция ошибок, она же функция Лапласа, он же интеграл вероятности — все это одна и та же сущность, которая выражается функцией

$\operatorname {erf}\,x={\frac {2}{{\sqrt {\pi }}}}\int \limits _{0}^{x}e^{{-t^{2}}}\,{\mathrm d}t$

и используется в статистике и теории вероятностей.

Функция неэлементарная, то есть её нельзя представить в виде элементарных (тригонометрических и алгебраических) функций.

Для расчета в нашем калькуляторе, мы используем связь с неполной гамма функцией

$\operatorname {erf}\,x=1-{\frac {\Gamma \left({\frac {1}{2}},x^{2}\right)}{{\sqrt \pi }}}$

Кроме этого мы сможем здесь же вычислить, дополнительную функцию ошибок, обозначаемую $\operatorname {erfc} \,x$ (иногда применяется обозначение $\operatorname {Erf} \,x$ ) и определяется через функцию ошибок:

$\operatorname {erfc}\,x=1-\operatorname {erf}\,x={\frac {2}{{\sqrt {\pi }}}}\int \limits _{x}^{{\infty }}e^{{-t^{2}}}\,{\mathrm d}t$

В приницпе это все, что можно сказать о ней.

Калькулятор высчитывает результат как в вещественном так и комплексном поле.

Замечание: Функция прекрасно работает на всем поле комплексных чисел при условии если аргумент ( фаза) меньше 180 градусов. Это связано с особенностью вычисления этой функции, неполной гамма функции, интегральной показательной функцией через непрерывные дроби.

Отсюда следует вывод, что при отрицательных вещественных аргументах, функция будет выдавать неверные решения. Но при всех положительных, а также отрицательных комплексных аргументах функция ошибок выдает верный ответ.

Несколько примеров:

Источник

3.3.Температурное
поле непрерывного неподвижного точечного
источника в неограниченной среде.
Функция ошибок Гаусса (функция erf(x)).

Если в точке с
координатами x‘,
y‘,
z‘
в интервале времени от t‘
= 0 до t‘
= t
работает источник тепла мощностью
W,
то температурное поле этого источника,
как указано выше, может быть найдено
интегрированием фундаментального
решения по t‘
от 0 до t
(т.е. от момента включения до момента
выключения источника). Поместим начало
координат в точку, где находится источник
тепла. Тогда x’
= y’
= z’
= 0, и формула
для температуры принимает вид:

,
(3.3.1)

где r²
= (x — x’)²
+ (y — y’)²
+ (z — z’)²
= x²
+ y²
+ z²
— квадрат расстояния от источника до
точки наблюдения.

Произведем в
интеграле (3.3.1) замену переменных:
r²/[4a(t
— t’)] = ².
Тогда: (t —
t’)^3/2
= r³/(8a^3/2³),
dt’ = r²d/(2a³),
пределы интегрирования: t’
= 0 
,
t’ = t


= ,
и формула (3.3.1) принимает вид:

.
(3.3.2)

Первый интеграл,
стоящий в скобках, известен из курса
высшей математики:

(интеграл
Пуассона),

а второй интеграл
через элементарные функции не выражается
и определяет специальную функцию,
которая называется функцией
ошибок Гаусса,
или интегралом
вероятностей,
или функцией эрфектум:

(3.3.3)

(читается «эрфектум»
или сокращенно: «эрф»). Через эту
функцию выражаются решения многих
задач в теории теплопроводности, да и
в других областях физики она играет
важную роль.

Из определения
(3.3.3) видно, что erf(0)
= 0, а erf()
= 1, т.е. erf(x)
— это монотонно возрастающая
функция, вид которой изображен
на Рис.3.3. Функция erf(x)
табулирована, и ее значения
приводятся в различных
справочниках; в таблице 3.1 приведены
несколько значений этой функции. В
библиотеках некоторых
языков программирования имеются
готовые подпрограммы для
вычисления функции erf(x).
Если готовой подпрограммы
нет, функцию erf(x)
можно
вычислить с помощью степенного
ряда. «Стандартное»
разложение этой функции в
степенной ряд, которое обычно
приводится в математических
справочниках, имеет вид:

.
(3.3.4)

Этот
ряд удобен для анализа свойств функции,
но для практических расчетов он неудобен,
т.к. является знакопеременным, что
при вычислениях приводит к потере
точности. Более удобен следующий
ряд:

,
(3.3.5)

где

Рис. 3.3.

помощью этого ряда легко составить
программу вычисления erf(x)
на любом языке программирования
и даже на программируемом
микрокалькуляторе. Суммирование
надо прекращать, когда при
добавлении очередного a_n-го
слагаемого сумма перестанет меняться
(будет достигнута «машинная
точность»).

Если большой
точности не требуется, то можно
использовать приближенную формулу:

erf(x)

[1 — exp(-4x²/)]^1/2.
(3.3.6)

Формула (3.3.6) дает
значения, абсолютная погрешность которых
не более 6.310^-3,
а относительная погрешность
не более 0.71%.

Иногда требуется
определить erf(x)
в области отрицательных значений x.
Из формулы (3.3.3) очевидно, что erf(-x)
= — erf(x).

Заметим, что хотя
функция erf(x)
не является «элементарной», с точки
зрения ее свойств и способов
вычисления она проще, чем многие
«элементарные» функции, например,
тригонометрические.

С функцией erf(x)
связано еще несколько функций, часто
встречающихся в теплофизических
задачах. Это прежде всего дополнительный
интеграл вероятностей:

,
(3.3.7)

который встречается
настолько часто, что для него используется
специальное обозначение: erfc(x)
(сокращенно читается «эрфик»). Вид
этой функции также приведен на рис.3.3.

Довольно часто
функцию erf(x)
приходится дифференцировать и
интегрировать. Из определения
(3.3.3) следует, что

,
(3.3.8)

а интеграл от
erfc(x)
(обозначается как ierfc(x))
равен:

.
(3.3.9)

Вернемся к формуле
(3.3.2). Замечая, что ca
= ,
запишем эту формулу в виде:

.
(3.3.10)

При t


значение функции

0,


1, и формула (3.3.10), как и должно быть,
совпадает с формулой для
стационарного решения (если T₀
принять за начало отсчета
температуры), т.к. при t


достигается стационарное
распределение температуры
в безграничной среде.

Таблица 3.1.
Некоторые значения функции erf(x).

x	erf(x)	x	erf(x)	x	erf(x)	x	erf(x)	x	erf(x)
0.0	0.0	0.3	0.32863	0.6	0.60386	0.9	0.79691	2.0	0.99532
0.1	0.11246	0.4	0.42839	0.7	0.67780	1.0	0.84270	2.5	0.99959
0.2	0.22270	0.5	0.52050	0.8	0.74210	1.5	0.96611

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Информация

Интеграл вероятности

Области знаний:: Интегральные преобразования и специальные функции
Другие наименования:: Интеграл ошибок

Интеграл вероятности

Интегра́л вероя́тности (интеграл ошибок), функция erf⁡(x)=2π∫0xe−t2 dt,∣x∣<∞.\operatorname{erf} (x)=\frac{2}{\sqrt{π}} \int\limits_0^x e^{-t^2}\, dt, \quad|x|<∞.В теории вероятностей обычно используется не интеграл вероятности, а функция распределения стандартного нормального закона Ф(x)=12π∫−∞xe−t2/2 dt=12(1+erf⁡(x/2)),\text{Ф}(x)=\frac{1}{\sqrt{2π}} \int\limits_{-\infty}^x e^{-t^2/2}\, dt=\frac{1}{2}(1+\operatorname{erf}(x/ \sqrt{2})),иногда называемая интегралом вероятности Гаусса. Для случайной величины XX, имеющей нормальное распределение с математическим ожиданием aa и дисперсией σ2σ^2, вероятность неравенства ∣(X−a)/σ∣⩽x|(X-a)/σ|⩽x равна erf (x/2)\text{erf}\,(x/\sqrt{2}).

Опубликовано 9 ноября 2022 г. в 12:21 (GMT+3). Последнее обновление 9 ноября 2022 г. в 12:21 (GMT+3).

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Таблицы DPVA.ru — Инженерный Справочник

Адрес этой страницы (вложенность) в справочнике dpva.ru: главная страница / / Техническая информация / / Математический справочник / / Теория вероятностей. Математическая статистика. Комбинаторика. / / Таблица. Интеграл вероятности или интеграл вероятностей. Таблица значений функции Лапласа. Она же функция ошибок erf

Таблица. Интеграл вероятности или интеграл вероятностей. Таблица значений функции Лапласа. Она же функция ошибок erf.

Интегральная функция вероятности распределения обычно выражается через специальную функцию erf(z).

z	0	1	2	3	4	5	6	7	8	9
0,00	0,0000	0011	0023	0034	0045	0056	0068	0079	0090	0102
1	0113	0124	0135	0147	0158	0169	0181	0192	0203	0214
2	0226	0237	0248	0260	0271	0282	0293	0305	0316	0327
3	0338	0350	0361	0372	0384	0395	0406	0417	0429	0553
4	0451	0462	0474	0485	0496	0507	0519	0530	0541	0553
5	0564	0575	0586	0598	0609	0620	0631	0643	0654	0665
6	0676	0688	0699	0710	0721	0732	0744	0755	0766	0777
7	0789	0800	0811	0822	0834	0845	0856	0867	0878	0890
8	0901	0912	0923	0934	0946	0957	0968	0979	0990	1002
9	1013	1024	1035	1046	1058	1069	1080	1091	1102	1113
10	1125	1136	1147	1158	1169	1180	1192	1203	1214	1225
1	1236	1247	1259	1270	1281	1292	1303	1314	1325	1386
2	1348	1359	1370	1381	1392	1403	1414	1425	1436	1448
3	1459	1470	1481	1492	1503	1514	1525	1536	1547	1558
4	1569	1581	1592	1603	1614	1625	1636	1647	1658	1669
5	1680	1691	1702	1713	1724	1735	1746	1757	1768	1779
6	1790	1801	1812	1823	1834	1845	1856	1867	1878	1889
7	1900	1911	1922	1933	1944	1955	1966	1977	1988	1998
8	2009	2020	2031	2042	2053	2064	2075	2086	2097	2108
9	2118	2129	2140	2151	2162	2173	2184	2194	2205	2216
20	2227	2238	2249	2260	2270	2281	2292	2303	2314	2324
1	2335	2346	2357	2368	2378	2389	2400	2411	2421	2432
2	2443	2454	2464	2475	2486	2497	2507	2518	2529	2540
3	2550	2561	2572	2582	2593	2604	2614	2625	2636	2646
4	2657	2668	2678	2689	2700	2710	2721	2731	2742	2753
5	2763	2774	2784	2795	2806	2816	2827	2837	2848	2858
6	2869	2880	2890	2901	2911	2922	2932	2943	2953	2964
7	2974	2985	2995	3006	3016	3027	3037	3047	3058	3068
8	3079	3089	3100	3110	3120	3131	3141	3152	3162	3276
9	3183	3193	3204	3214	3224	3235	3246	3255	3266	3276
30	3286	3297	3307	3317	3327	3338	3348	3358	3369	3379
1	3389	3399	3410	3420	3430	3440	3450	3461	3471	3481
2	3491	3501	3512	3522	3532	3542	3552	3562	3573	3583
3	3593	3603	3613	3623	3633	3643	3653	3663	3674	3684
4	3694	3704	3714	3724	8734	3744	2754	3764	3774	3784
5	3794	3804	3814	3824	3834	3844	3854	3864	3873	3883
6	3893	3903	3913	3923	3933	8943	3953	3963	3972	3982
7	3992	4002	4012	4022	4031	4041	4051	4061	4071	4080
8	4090	4100	4110	4119	4129	4189	4149	4158	4168	4178
9	4187	4197	4207	4216	4226	4236	4245	4255	4265	4274
40	4284	4294	4303	4313	4322	4332	4341	4351	4361	4370
1	4380	4389	4399	4408	4418	4427	4437	4446	4456	4465
2	4475	4484	4494	4503	4512	4522	4531	4541	4550	4559
3	4569	4578	4588	4597	4606	4616	4625	4634	4644	4653
4	4662	4672	4681	4690	4699	4709	4718	4727	4736	4746
5	4755	4764	4773	4782	4792	4801	4810	4819	4828	4837
6	4847	4856	4865	4874	4883	4892	4901	4910	4919	4928
7	4937	4946	4956	4965	4974	4983	4992	5001	5010	5019
8	5027	5036	5045	5054	5063	5072	5081	5090	5099	5108
9	5117	5126	5134	5143	5152	5161	5170	5179	5187	5196

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Источник

Name[edit]

Applications[edit]

Properties[edit]

Taylor series[edit]

Derivative and integral[edit]

Bürmann series[edit]

Inverse functions[edit]

Asymptotic expansion[edit]

Continued fraction expansion[edit]

Integral of error function with Gaussian density function[edit]

Factorial series[edit]

Numerical approximations[edit]

Approximation with elementary functions[edit]

Table of values[edit]

[edit]

Complementary error function[edit]

Imaginary error function[edit]

Cumulative distribution function[edit]

Generalized error functions[edit]

Iterated integrals of the complementary error function[edit]

Implementations[edit]

As real function of a real argument[edit]

As complex function of a complex argument[edit]

See also[edit]

[edit]

In probability[edit]

References[edit]

Further reading[edit]

External links[edit]

Функция ошибок

Интеграл вероятности

Интеграл вероятности

Таблица. Интеграл вероятности или интеграл вероятностей. Таблица значений функции Лапласа. Она же функция ошибок erf.