Erf: Error function—Wolfram Documentation (original) (raw)

Function Identities and Simplifications

Generalizations & Extensions

Erf[z]

gives the error function .

Erf[z0,z1]

gives the generalized error function .

Details

Examples

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Basic Examples (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope (40)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate Erf efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix Erf function using MatrixFunction:

Specific Values (3)

Simple exact values are generated automatically:

Values at infinity:

Find the zero of Erf:

Visualization (2)

Plot the Erf function:

Plot the real part of :

Plot the imaginary part of :

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(TemplateBox[{z}, Conjugate])=TemplateBox[{{erf, (, z, )}}, Conjugate]:

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:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the n^(th) derivative:

Integration (3)

Indefinite integral of Erf:

Definite integral of an odd integrand over an interval centered at the origin is 0:

More integrals:

Series Expansions (4)

Taylor expansion for Erf:

Plot the first three approximations for Erf around :

General term in the series expansion of Erf:

Asymptotic expansion of Erf:

Erf can be applied to a power series:

Integral Transforms (2)

Function Identities and Simplifications (3)

Integral definition of the error function:

Argument involving basic arithmetic operations:

The two-argument form gives the difference:

Function Representations (4)

Generalizations & Extensions (1)

The two-argument form gives the difference:

Applications (3)

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:

The plot of the solution for different times:

Under an excess of loss reinsurance agreement, a claim is shared between the insurer and reinsurer only if the claim exceeds a fixed amount, called the retention level. Otherwise, the insurer pays the claim in full. Compute the expected value of the amounts and , paid by the insurer and the reinsurer for a retention level of if the claims follow a lognormal distribution with parameters and . Find the expected insurer claim payouts:

Find the expected reinsurer payouts to the insurer:

Properties & Relations (3)

Compose with inverse functions:

Solve a transcendental equation:

Erf appears in special cases of many mathematical functions:

Possible Issues (3)

For large arguments, intermediate values may underflow:

The error function for large real-part arguments can be very close to 1:

Very large arguments can give unevaluated results:

Neat Examples (2)

Plot a clothoid:

A continued fraction whose partial numerators are consecutive integers:

Its limit can be expressed in terms of Erf:

▪

Wolfram Research (1988), Erf, Wolfram Language function, https://reference.wolfram.com/language/ref/Erf.html (updated 2022).

Text

Wolfram Research (1988), Erf, Wolfram Language function, https://reference.wolfram.com/language/ref/Erf.html (updated 2022).

CMS

Wolfram Language. 1988. "Erf." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Erf.html.

APA

Wolfram Language. (1988). Erf. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Erf.html

BibTeX

@misc{reference.wolfram_2025_erf, author="Wolfram Research", title="{Erf}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Erf.html}", note=[Accessed: 20-October-2025]}

BibLaTeX

@online{reference.wolfram_2025_erf, organization={Wolfram Research}, title={Erf}, year={2022}, url={https://reference.wolfram.com/language/ref/Erf.html}, note=[Accessed: 20-October-2025]}