Error function

Error function
Plot of the error function over real numbers
General information
General definition	ℂ
Basic features
Parity	Odd
Specific features
Root	0
Derivative	erf⁡(z)=2π∑n=0∞(−1)nz2n+1n!(2n+1)=2π(z−z33+z510−z742+z9216⋯)

In mathematics, the error function (also called the Gauss error function), often denoted by exp⁡(−t2) is holomorphic on the whole complex plane X, the error function has the following interpretation: for a real random variable erf⁡(x) is the probability that erfc⁡(z)=1−erf⁡(z)

and the imaginary error function

Name

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The name "error function" and its abbreviation p and σ and expected value zero, then

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 normally distributed random variable σ and a constant \\mu"}}'>L>μ, it can be shown via integration by substitution that

B are certain numeric constants. If Pr⁡[X≤L]≤Aexp⁡(−Blog⁡(k))=AkB

and so the probability goes to 0 as [La,Lb] can be derived as e−t2 is an even function (since 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 maps real numbers to real numbers, for any complex number z¯ denotes the complex conjugate of 1 (see Gaussian integral). At the real axis, z→∞ and ±i∞.

Taylor series

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The error function is an entire function; it has no singularities (except at infinity) and its Taylor expansion always converges. For erf⁡(0)=0, one obtains the error function's Maclaurin series as: −(2k−1)z2k(2k+1)

expresses the multiplier to turn the z as the first term).

The imaginary error function has a similar Maclaurin series: ddzerf⁡(z)=2πe−z2. From this, the derivative of the imaginary error function is also immediate: Hk are the physicists' Hermite polynomials.

An antiderivative of the error function, obtainable by integration by parts, is x than a Taylor expansion is obtained by using Bürmann's theorem: c1=31/200 and 0.0034361: w satisfying erf−1(x) satisfying

|z|<1 of the complex plane, using the Maclaurin series

ck=∑m=0k−1cmck−1−m(m+1)(2m+1)={1,1,76,12790,43692520,3480716200,…}.

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

(sequence A092677 in the OEIS) respectively; without cancellation the numerator terms are values in ±1.

For erfc−1(1−z)=erf−1(z). For real erfi⁡(erfi−1(x))=x. The inverse imaginary error function is defined as erfi−1(z)=∑k=0∞(−1)kck2k+1(π2z)2k+1,

where erfc⁡(x)=e−x2xπ(1+∑n=1∞(−1)n1⋅3⋅5⋯(2n−1)(2x2)n)=e−x2xπ∑n=0∞(−1)n(2n−1)!!(2x2)n, where 2n−1. This series diverges for every finite erfc⁡(x)=e−x2xπ∑n=0N−1(−1)n(2n−1)!!(2x2)n+RN(x),

where the remainder is

RN(x)=O(x−(1+2N)e−x2)

as x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc⁡(z)=zπe−z21z2+a11+a2z2+a31+⋯ where Re⁡(z2)>0. Here

s(n,k) denotes a signed Stirling number of the first kind. The Taylor series can be written in terms of the double factorial:

5×10)

a2 = 0.230389, 3</sub> {{=}} 0.000972"}},"i":0}}]}'>a3 = 0.000972, 4</sub> {{=}} 0.078108"}},"i":0}}]}'>a4 = 0.078108

where a2 = −0.0958798, 3</sub> {{=}} 0.7478556"}},"i":0}}]}'>a3 = 0.7478556