Built-in Mathematica Symbol

Erf

Erf[z]
gives the error function

.

Erf[z₀, z₁]
gives the generalized error function

.

MORE INFORMATION

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

CLOSE ALL

Basic Examples (3)

Evaluate numerically:

Scope (6)

Evaluate for complex arguments:

Evaluate to high precision:

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

Erf threads element-wise over lists:

Simple exact values are generated automatically:

Parity transformation is automatically applied:

Generalizations & Extensions (3)

The two-argument form gives the difference:

Erf can be applied to a power series:

Infinite arguments give symbolic results:

Applications (2)

CDF of normal distribution:

Cumulative probabilities for values of normal random variable to lie between -n

and n

:

Solution of the heat equation for piecewise-constant initial condition:

A check that the solution fulfills the heat equation:

Plot of the solution for different times:

Properties & Relations (5)

Compose with inverse functions:

Solve a transcendental equation:

Integrals:

Integral transforms:

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: