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.
Erf can be used with Interval and CenteredInterval objects. »

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:

Erf threads elementwise over lists:

Erf can be used with Interval and CenteredInterval objects:

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 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 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:

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Erf

Details

Examples

Basic Examples (5)