Erf

Erf[z]

gives the error function .

Erf[z₀,z₁]

gives the generalized error function .

Details

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.

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 (33)

Numerical Evaluation (5)

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:

Erf threads elementwise over lists:

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 (4)

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 :

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the 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)

Compute the Fourier transform of Erf using FourierTransform:

LaplaceTransform:

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)

Error function in terms of the incomplete Gamma:

Represent in terms of MeijerG using MeijerGReduce:

Erf can be represented as a DifferentialRoot:

TraditionalForm formatting:

Generalizations & Extensions (1)

The two-argument form gives the difference:

Applications (2)

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:

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 (1)

A neat continued fraction:

Its limit can be expressed through Erf:

Introduced in 1988

(1.0)

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Erf