- Mathematical function, suitable for both symbolic and numerical manipulation.
- Erf[z] is the integral of the Gaussian distribution, given by .
- Erf[z0,z1] is given by .
- Erf[z] is an entire function of z with no branch cut discontinuities.
- For certain special arguments, Erf automatically evaluates to exact values.
- Erf can be evaluated to arbitrary numerical precision.
- Erf automatically threads over lists.
- Erf can be used with Interval and CenteredInterval objects. »
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Basic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate Erf efficiently at high precision:
Erf threads elementwise over lists:
Erf can be used with Interval and CenteredInterval objects:
Specific Values (3)
Simple exact values are generated automatically:
Find the zero of Erf:
Plot the Erf function:
Function Properties (10)
Erf is defined for all real and complex values:
Erf takes all real values between –1 and 1:
Erf is an odd function:
Erf has the mirror property :
Erf is an analytic function of x:
It has no singularities or discontinuities:
Erf is nondecreasing:
Erf is injective:
Erf is not surjective:
Erf is neither non-negative nor non-positive:
Erf is neither convex nor concave:
Indefinite integral of Erf:
Definite integral of an odd integrand over an interval centered at the origin is 0:
Series Expansions (4)
Integral Transforms (2)
Compute the Fourier transform of Erf using FourierTransform:
Function Identities and Simplifications (3)
Function Representations (4)
Error function in terms of the incomplete Gamma:
Represent in terms of MeijerG using MeijerGReduce:
Erf can be represented as a DifferentialRoot:
Express the CDF of NormalDistribution in terms of the error function:
The cumulative probabilities for values of the normal random variable lie between -n σ and n σ:
The solution of the heat equation for a piecewise‐constant initial condition:
A check that the solution fulfills the heat equation:
Properties & Relations (3)
Compose with inverse functions:
Solve a transcendental equation:
Erf appears in special cases of many mathematical functions:
Possible Issues (3)
Neat Examples (1)
Its limit can be expressed through Erf:
Wolfram Research (1988), Erf, Wolfram Language function, https://reference.wolfram.com/language/ref/Erf.html (updated 2022).
Wolfram Language. 1988. "Erf." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Erf.html.
Wolfram Language. (1988). Erf. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Erf.html