Error function: Difference between revisions
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Template:Short description Template:Use dmy dates Template:Distinguish In mathematics, the error function (also called the Gauss error function), often denoted by Template:Math, is a function <math>\mathrm{erf}: \mathbb{C} \to \mathbb{C}</math> defined as:<ref>Template:Cite book</ref> <math display="block">\operatorname{erf} z = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,\mathrm dt.</math> Template:Infobox mathematical function{\sqrt\pi} + C</math> | taylor_series = <math>\operatorname{erf} z = \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} \frac{z^{2n+1}}{n!}</math> }}
The integral here is a complex contour integral which is path-independent because <math>\exp(-t^2)</math> is holomorphic on the whole complex plane <math>\mathbb{C}</math>. In many applications, the function argument is a real number, in which case the function value is also real.
In some old texts,<ref>Template:Cite book</ref> the error function is defined without the factor of <math>\frac{2}{\sqrt{\pi}}</math>. This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations.
In statistics, for non-negative real values of Template:Mvar, the error function has the following interpretation: for a real random variable Template:Mvar that is normally distributed with mean 0 and standard deviation <math>\frac{1}{\sqrt{2}}</math>, Template:Math is the probability that Template:Mvar falls in the range Template:Closed-closed.
Two closely related functions are the complementary error function <math>\mathrm{erfc}: \mathbb{C} \to \mathbb{C}</math> is defined as
<math display="block">\operatorname{erfc} z = 1 - \operatorname{erf} z,</math>
and the imaginary error function <math>\mathrm{erfi}: \mathbb{C} \to \mathbb{C}</math> is defined as
<math display="block">\operatorname{erfi} z = -i\operatorname{erf} iz,</math>
where Template:Mvar is the imaginary unit.
Name
[edit]The name "error function" and its abbreviation Template:Math 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."<ref name="Glaisher1871a">Template:Cite journal</ref> The error function complement was also discussed by Glaisher in a separate publication in the same year.<ref name="Glaisher1871b">Template:Cite journal</ref> For the "law of facility" of errors whose density is given by <math display="block">f(x) = \left(\frac{c}{\pi}\right)^{1/2} e^{-c x^2}</math> (the normal distribution), Glaisher calculates the probability of an error lying between Template:Mvar and Template:Mvar as: <math display="block">\left(\frac{c}{\pi}\right)^\frac{1}{2} \int_p^qe^{-cx^2}\,\mathrm dx = \tfrac{1}{2}\left(\operatorname{erf} \left(q\sqrt{c}\right) -\operatorname{erf} \left(p\sqrt{c}\right)\right).</math>
_in_the_complex_plane_from_-2-2i_to_2%2B2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg){kind=link}
Applications
[edit]When the results of a series of measurements are described by a normal distribution with standard deviation Template:Mvar and expected value 0, then Template:Math is the probability that the error of a single measurement lies between Template:Math and Template:Math, for positive Template:Mvar. 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 Template:Math (a normal distribution with mean Template:Mvar and standard deviation Template:Mvar) and a constant Template:Math, it can be shown via integration by substitution: <math display="block">\begin{align} \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{align}</math>
where Template:Mvar and Template:Mvar are certain numeric constants. If Template:Mvar is sufficiently far from the mean, specifically Template:Math, then:
<math display="block">\Pr[X\leq L] \leq A \exp (-B \ln{k}) = \frac{A}{k^B}</math>
so the probability goes to 0 as Template:Math.
The probability for Template:Mvar being in the interval Template:Closed-closed can be derived as <math display="block">\begin{align} \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 dx \\ &= \frac{1}{2}\left(\operatorname{erf}\frac{L_b-\mu}{\sqrt{2}\sigma} - \operatorname{erf}\frac{L_a-\mu}{\sqrt{2}\sigma}\right).\end{align}</math>
Properties
[edit]The property Template:Math means that the error function is an odd function. This directly results from the fact that the integrand Template:Math 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 Template:Mvar: <math display="block">\operatorname{erf} \overline{z} = \overline{\operatorname{erf} z} </math> where <math>\overline{z} </math> denotes the complex conjugate of <math>z</math>.
The integrand Template:Math and Template:Math are shown in the complex Template:Mvar-plane in the figures at right with domain coloring.
The error function at Template:Math is exactly 1 (see Gaussian integral). At the real axis, Template:Math approaches unity at Template:Math and −1 at Template:Math. At the imaginary axis, it tends to Template:Math.
Taylor series
[edit]The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges. For Template:Math, however, cancellation of leading terms makes the Taylor expansion unpractical.
The defining integral cannot be evaluated in closed form in terms of elementary functions (see Liouville's theorem), but by expanding the integrand Template:Math into its Maclaurin series and integrating term by term, one obtains the error function's Maclaurin series as: <math display="block">\begin{align} \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{align}</math> which holds for every complex number Template:Mvar. The denominator terms are sequence A007680 in the OEIS.
For iterative calculation of the above series, the following alternative formulation may be useful: <math display="block">\begin{align} \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{align}</math> because Template:Math expresses the multiplier to turn the Template:Mvarth term into the Template:Mathth term (considering Template:Mvar as the first term).
The imaginary error function has a very similar Maclaurin series, which is: <math display="block">\begin{align} \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{align}</math> which holds for every complex number Template:Mvar.
Derivative and integral
[edit]The derivative of the error function follows immediately from its definition: <math display="block">\frac{\mathrm d}{\mathrm dz}\operatorname{erf} z =\frac{2}{\sqrt\pi} e^{-z^2}.</math> From this, the derivative of the imaginary error function is also immediate: <math display="block">\frac{d}{dz}\operatorname{erfi} z =\frac{2}{\sqrt\pi} e^{z^2}.</math> An antiderivative of the error function, obtainable by integration by parts, is <math display="block">z\operatorname{erf}z + \frac{e^{-z^2}}{\sqrt\pi}+C.</math> An antiderivative of the imaginary error function, also obtainable by integration by parts, is <math display="block">z\operatorname{erfi}z - \frac{e^{z^2}}{\sqrt\pi}+C.</math> Higher order derivatives are given by <math display="block">\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 dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots</math> where Template:Mvar are the physicists' Hermite polynomials.<ref>Template:Mathworld</ref>
Bürmann series
[edit]An expansion,<ref>Template:Cite journal</ref> which converges more rapidly for all real values of Template:Mvar than a Taylor expansion, is obtained by using Hans Heinrich Bürmann's theorem:<ref>Template:Mathworld</ref> <math display="block">\begin{align} \operatorname{erf} x &= \frac{2}{\sqrt\pi} \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}{276 480} \left (1-e^{-x^2} \right )^4 - \cdots \right) \\[10pt] &= \frac{2}{\sqrt\pi} \sgn x \cdot \sqrt{1-e^{-x^2}} \left(\frac{\sqrt\pi}{2} + \sum_{k=1}^\infty c_k e^{-kx^2} \right). \end{align}</math> where Template:Math is the sign function. By keeping only the first two coefficients and choosing Template:Math and Template:Math, the resulting approximation shows its largest relative error at Template:Math, where it is less than 0.0034361: <math display="block">\operatorname{erf} x \approx \frac{2}{\sqrt\pi}\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). </math>
Inverse functions
[edit]
Given a complex number Template:Mvar, there is not a unique complex number Template:Mvar satisfying Template:Math, so a true inverse function would be multivalued. However, for Template:Math, there is a unique real number denoted Template:Math satisfying <math display="block">\operatorname{erf}\left(\operatorname{erf}^{-1} x\right) = x.</math>
The inverse error function is usually defined with domain Template:Open-open, and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk Template:Math of the complex plane, using the Maclaurin series<ref>Template:Cite arXiv</ref> <math display="block">\operatorname{erf}^{-1} z=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt\pi}{2}z\right )^{2k+1},</math> where Template:Math and <math display="block">\begin{align} 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{align}</math>
So we have the series expansion (common factors have been canceled from numerators and denominators): <math display="block">\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 ).</math> (After cancellation the numerator and denominator values in Template:Oeis and Template:Oeis respectively; without cancellation the numerator terms are values in Template:Oeis.) The error function's value at Template:Math is equal to Template:Math.
For Template:Math, we have Template:Math.
The inverse complementary error function is defined as <math display="block">\operatorname{erfc}^{-1}(1-z) = \operatorname{erf}^{-1} z.</math> For real Template:Mvar, there is a unique real number Template:Math satisfying Template:Math. The inverse imaginary error function is defined as Template:Math.<ref>Template:Cite arXiv</ref>
For any real x, Newton's method can be used to compute Template:Math, and for Template:Math, the following Maclaurin series converges: <math display="block">\operatorname{erfi}^{-1} z =\sum_{k=0}^\infty\frac{(-1)^k c_k}{2k+1} \left( \frac{\sqrt\pi}{2} z \right)^{2k+1},</math> where Template:Math 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 Template:Mvar is <math display="block">\begin{align} \operatorname{erfc} x &= \frac{e^{-x^2}}{x\sqrt{\pi}}\left(1 + \sum_{n=1}^\infty (-1)^n \frac{1\cdot3\cdot5\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{align}</math> where Template:Math is the double factorial of Template:Math, which is the product of all odd numbers up to Template:Math. This series diverges for every finite Template:Mvar, and its meaning as asymptotic expansion is that for any integer Template:Math one has <math display="block">\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)</math> where the remainder is <math display="block">R_N(x) := \frac{(-1)^N \, (2 N - 1)!!}{\sqrt{\pi} \cdot 2^{N - 1}} \int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt,</math> which follows easily by induction, writing <math display="block">e^{-t^2} = -\frac{1}{2 t} \, \frac{\mathrm{d}}{\mathrm{d}t} e^{-t^2}</math> and integrating by parts.
The asymptotic behavior of the remainder term, in Landau notation, is <math display="block">R_N(x) = O\left(x^{- (1 + 2N)} e^{-x^2}\right)</math> as Template:Math. This can be found by <math display="block">R_N(x) \propto \int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt = e^{-x^2} \int_0^\infty (t+x)^{-2N}e^{-t^2-2tx}\,\mathrm dt\leq e^{-x^2} \int_0^\infty x^{-2N} e^{-2tx}\,\mathrm dt \propto x^{-(1+2N)}e^{-x^2}.</math> For large enough values of Template:Mvar, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of Template:Math (while for not too large values of Template:Mvar, the above Taylor expansion at 0 provides a very fast convergence).
Continued fraction expansion
[edit]A continued fraction expansion of the complementary error function was found by Laplace:<ref>Pierre-Simon Laplace, Traité de mécanique céleste, tome 4 (1805), livre X, page 255.</ref><ref>Template:Cite book</ref> <math display="block">\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}.</math>
Factorial series
[edit]The inverse factorial series: <math display="block">\begin{align} \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{align}</math> converges for Template:Math. Here <math display="block">\begin{align} 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{align}</math> Template:Math denotes the rising factorial, and Template:Math denotes a signed Stirling number of the first kind.<ref>Template:Cite journal</ref><ref>Template:Cite book</ref> There also exists a representation by an infinite sum containing the double factorial: <math display="block">\operatorname{erf} z = \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{(-2)^n(2n-1)!!}{(2n+1)!}z^{2n+1}</math>
Bounds and 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: <math display="block">\operatorname{erf} x \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + a_3x^3 + a_4x^4\right)^4}, \qquad x \geq 0</math> (maximum error: Template:Val) Template:Pb where Template:Math, Template:Math, Template:Math, Template:Math <math display="block">\operatorname{erf} x \approx 1 - \left(a_1t + a_2t^2 + a_3t^3\right)e^{-x^2},\quad t=\frac{1}{1 + px}, \qquad x \geq 0</math> (maximum error: Template:Val) Template:Pb where Template:Math, Template:Math, Template:Math, Template:Math <math display="block">\operatorname{erf} x \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + \cdots + a_6x^6\right)^{16}}, \qquad x \geq 0</math> (maximum error: Template:Val) Template:Pb where Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math <math display="block">\operatorname{erf} x \approx 1 - \left(a_1t + a_2t^2 + \cdots + a_5t^5\right)e^{-x^2},\quad t = \frac{1}{1 + px}</math> (maximum error: Template:Val) Template:Pb where Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math Template:Pb All of these approximations are valid for Template:Math. To use these approximations for negative Template:Mvar, use the fact that Template:Math is an odd function, so Template:Math.
- Exponential bounds and a pure exponential approximation for the complementary error function are given by<ref>Template:Cite journal</ref> <math display="block">\begin{align} \operatorname{erfc} x &\leq \frac{1}{2}e^{-2 x^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{align}</math>
- The above have been generalized to sums of Template:Mvar exponentials<ref>Template:Cite journal</ref> with increasing accuracy in terms of Template:Mvar so that Template:Math can be accurately approximated or bounded by Template:Math, where <math display="block">\tilde{Q}(x) = \sum_{n=1}^N a_n e^{-b_n x^2}.</math> In particular, there is a systematic methodology to solve the numerical coefficients Template:Math that yield a minimax approximation or bound for the closely related Q-function: Template:Math, Template:Math, or Template:Math for Template:Math. The coefficients Template:Math for many variations of the exponential approximations and bounds up to Template:Math have been released to open access as a comprehensive dataset.<ref>Template:Cite journal</ref>
- A tight approximation of the complementary error function for Template:Math is given by Karagiannidis & Lioumpas (2007)<ref>Template:Cite journal</ref> who showed for the appropriate choice of parameters Template:Math that <math display="block">\operatorname{erfc} x \approx \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x}.</math> They determined Template:Math, which gave a good approximation for all Template:Math. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.<ref>Template:Cite journal</ref>
- A single-term lower bound is<ref>Template:Cite journal</ref> <math display="block" display="block">\operatorname{erfc} x \geq \sqrt{\frac{2 e}{\pi}} \frac{\sqrt{\beta - 1}}{\beta} e^{- \beta x^2}, \qquad x \ge 0,\quad \beta > 1,</math> where the parameter Template:Mvar can be picked to minimize error on the desired interval of approximation.
- Another approximation is given by Sergei Winitzki using his "global Padé approximations":<ref>Template:Cite book</ref><ref>Template:Cite journal</ref>Template:Rp <math display="block">\operatorname{erf} x \approx \sgn x \cdot \sqrt{1 - \exp\left(-x^2\frac{\frac{4}{\pi} + ax^2}{1 + ax^2}\right)}</math> where <math display="block">a = \frac{8(\pi - 3)}{3\pi(4 - \pi)} \approx 0.140012.</math> 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 Template:Mvar. Using the alternate value Template:Math reduces the maximum relative error to about 0.00013.<ref>Template:Cite web</ref> Template:Pb This approximation can be inverted to obtain an approximation for the inverse error function: <math display="block">\operatorname{erf}^{-1}x \approx \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)}.</math>
- An approximation with a maximal error of Template:Val for any real argument is:<ref>Template:Cite book</ref> <math display="block">\operatorname{erf} x = \begin{cases} 1-\tau & x\ge 0\\ \tau-1 & x < 0 \end{cases}</math> with <math display="block">\begin{align} \tau &= t\cdot\exp\left(-x^2-1.26551223+1.00002368 t+0.37409196 t^2+0.09678418 t^3 -0.18628806 t^4\right.\\ &\left. \qquad\qquad\qquad +0.27886807 t^5-1.13520398 t^6+1.48851587 t^7 -0.82215223 t^8+0.17087277 t^9\right) \end{align}</math> and <math display="block">t = \frac{1}{1 + \frac{1}{2}|x|}.</math>
- An approximation of <math>\operatorname{erfc}</math> with a maximum relative error less than <math>2^{-53}</math> <math>\left(\approx 1.1 \times 10^{-16}\right)</math> in absolute value is:<ref>Template:Cite journal</ref> for Template:Nowrap <math display="block">\begin{aligned} \operatorname{erfc} \left(x\right) & = \left(\frac{0.56418958354775629}{x+2.06955023132914151}\right) \left(\frac{x^2+2.71078540045147805 x+5.80755613130301624}{x^2+3.47954057099518960 x+12.06166887286239555}\right) \\ & \left(\frac{x^2+3.47469513777439592 x+12.07402036406381411}{x^2+3.72068443960225092 x+8.44319781003968454}\right) \left(\frac{x^2+4.00561509202259545 x+9.30596659485887898}{x^2+3.90225704029924078 x+6.36161630953880464}\right) \\ & \left(\frac{x^2+5.16722705817812584 x+9.12661617673673262}{x^2+4.03296893109262491 x+5.13578530585681539}\right) \left(\frac{x^2+5.95908795446633271 x+9.19435612886969243}{x^2+4.11240942957450885 x+4.48640329523408675}\right) e^{-x^2} \\ \end{aligned}</math> and for <math>x<0</math> <math display="block">\operatorname{erfc} \left(x\right) = 2 - \operatorname{erfc} \left(-x\right)</math>
- A simple approximation for real-valued arguments could be done through Hyperbolic functions: <math display="block">\operatorname{erf} \left(x\right) \approx z(x) = \tanh\left(\frac{2}{\sqrt{\pi}}\left(x+\frac{11}{123}x^3\right)\right)</math> which keeps the absolute difference Template:Nowrap
- Since the error function and the Gaussian Q-function are closely related through the identity <math>\operatorname{erfc}(x) = 2 Q(\sqrt{2} x)</math> or equivalently <math>Q(x) = \frac{1}{2} \operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right)</math>, 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 <math>x \in [0, \infty)</math> was introduced by Abreu (2012)<ref>Template:Cite journal</ref> based on a simple algebraic expression with only two exponential terms: <math display="block">Q(x) \geq \frac{1}{12} e^{-x^2} + \frac{1}{\sqrt{2\pi} (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0,</math> and <math display="block">Q(x) \leq \frac{1}{50} e^{-x^2} + \frac{1}{2 (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0.</math> These bounds stem from a unified form <math display="block">Q_{\mathrm{B}}(x; a, b) = \frac{\exp(-x^2)}{a} + \frac{\exp(-x^2 / 2)}{b (x + 1)},</math> where the parameters <math>a</math> and <math>b</math> are selected to ensure the bounding properties: for the lower bound, <math>a_{\mathrm{L}} = 12</math> and <math>b_{\mathrm{L}} = \sqrt{2\pi}</math>, and for the upper bound, <math>a_{\mathrm{U}} = 50</math> and <math>b_{\mathrm{U}} = 2</math>. 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 <math>Q^n(x)</math> for positive integers <math>n</math> via the binomial theorem, suggesting potential adaptability for powers of <math>\operatorname{erfc}(x)</math>, though this is less commonly required in error function applications.
Table of values
[edit]Related functions
[edit]Complementary error function
[edit]The complementary error function, denoted Template:Math, is defined as
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<math display="block">\begin{align} \operatorname{erfc} x & = 1-\operatorname{erf} x \\[5pt] & = \frac{2}{\sqrt\pi} \int_x^\infty e^{-t^2}\,\mathrm dt \\[5pt] & = e^{-x^2} \operatorname{erfcx} x, \end{align} </math> which also defines Template:Math, the scaled complementary error function<ref name=Cody93>Template:Citation</ref> (which can be used instead of Template:Math to avoid arithmetic underflow<ref name=Cody93/><ref name=Zaghloul07>Template:Citation</ref>). Another form of Template:Math for Template:Math is known as Craig's formula, after its discoverer:<ref>John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations Template:Webarchive, Proceedings of the 1991 IEEE Military Communication Conference, vol. 2, pp. 571–575.</ref> <math display="block">\operatorname{erfc} (x \mid x\ge 0) = \frac{2}{\pi} \int_0^\frac{\pi}{2} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) \, \mathrm d\theta.</math> This expression is valid only for positive values of Template:Mvar, but it can be used in conjunction with Template:Math to obtain Template:Math for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for the Template:Math of the sum of two non-negative variables is as follows:<ref>Template:Cite journal</ref> <math display="block">\operatorname{erfc} (x+y \mid x,y\ge 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.</math>
Imaginary error function
[edit]The imaginary error function, denoted Template:Math, is defined as
_in_the_complex_plane_from_-2-2i_to_2%2B2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg){kind=link}
<math display="block">\begin{align} \operatorname{erfi} x & = -i\operatorname{erf} ix \\[5pt] & = \frac{2}{\sqrt\pi} \int_0^x e^{t^2}\,\mathrm dt \\[5pt] & = \frac{2}{\sqrt\pi} e^{x^2} D(x), \end{align} </math> where Template:Math is the Dawson function (which can be used instead of Template:Math to avoid arithmetic overflow<ref name=Cody93/>).
Despite the name "imaginary error function", Template:Math is real when Template:Mvar is real.
When the error function is evaluated for arbitrary complex arguments Template:Mvar, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: <math display="block">w(z) = e^{-z^2}\operatorname{erfc}(-iz) = \operatorname{erfcx}(-iz).</math>
Cumulative distribution function
[edit]The error function is essentially identical to the standard normal cumulative distribution function, denoted Template:Math, also named Template:Math by some software languagesTemplate:Citation needed, as they differ only by scaling and translation. Indeed,
<math display="block">\begin{align} \Phi(x) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^\tfrac{-t^2}{2}\,\mathrm dt\\[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{align}</math> or rearranged for Template:Math and Template:Math: <math display="block">\begin{align}
\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{align}</math>
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 <math display="block">\begin{align} Q(x) &= \frac{1}{2} - \frac{1}{2} \operatorname{erf} \frac{x}{\sqrt 2}\\ &= \frac{1}{2}\operatorname{erfc}\frac{x}{\sqrt 2}. \end{align}</math>
The inverse of Template:Math is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as <math display="block">\operatorname{probit}(p) = \Phi^{-1}(p) = \sqrt{2}\operatorname{erf}^{-1}(2p-1) = -\sqrt{2}\operatorname{erfc}^{-1}(2p).</math>
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): <math display="block">\operatorname{erf} x = \frac{2x}{\sqrt\pi} M\left(\tfrac{1}{2},\tfrac{3}{2},-x^2\right).</math>
It has a simple expression in terms of the Fresnel integral.Template:Elucidate
In terms of the regularized gamma function Template:Mvar and the incomplete gamma function, <math display="block">\operatorname{erf} x = \sgn x \cdot P\left(\tfrac{1}{2}, x^2\right) = \frac{\sgn x}{\sqrt\pi} \gamma{\left(\tfrac{1}{2}, x^2\right)}.</math>Template:Math is the sign function.
Iterated integrals of the complementary error function
[edit]The iterated integrals of the complementary error function are defined by<ref>Template:Cite book</ref> <math display="block">\begin{align} 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 -2 z \operatorname{ierfc} z \right) \\ \end{align}</math>
The general recurrence formula is <math display="block">2 n \cdot i^n\!\operatorname{erfc} z = i^{n-2}\!\operatorname{erfc} z -2 z \cdot i^{n-1}\!\operatorname{erfc} z</math>
They have the power series <math display="block">i^n\!\operatorname{erfc} z =\sum_{j=0}^\infty \frac{(-z)^j}{2^{n-j}j! \,\Gamma \left( 1 + \frac{n-j}{2}\right)},</math> from which follow the symmetry properties <math display="block">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)!}</math> and <math display="block">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)!}. </math>
Implementations
[edit]As real function of a real argument
[edit]- In POSIX-compliant operating systems, the header
math.hshall declare and the mathematical librarylibmshall provide the functionserfanderfc(double precision) as well as their single precision and extended precision counterpartserff,erflanderfcf,erfcl.<ref>Template:Cite web</ref> - The GNU Scientific Library provides
erf,erfc,log(erf), and scaled error functions.<ref>Template:Cite web</ref>
As complex function of a complex argument
[edit]libcerf, numeric C library for complex error functions, provides the complex functionscerf,cerfc,cerfcxand the real functionserfi,erfcxwith approximately 13–14 digits precision, based on the Faddeeva function as implemented in the MIT Faddeeva Package