Skip to content

Erf

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Erf[z]
    gives the error function erf(z) = (2/Sqrt[Pi]) Integral_0^z e^(-t^2) dt.
Erf[z0, z1]
    gives the generalized error function erf(z1) - erf(z0).
Erf[0] = 0, Erf[Infinity] = 1, Erf[-Infinity] = -1. An entire function,
odd in z. Real and complex inputs evaluate numerically at machine or
arbitrary (MPFR) precision; D[Erf[z], z] = (2/Sqrt[Pi]) E^(-z^2). Listable.

Examples

No verified examples yet for this function.

Implementation notes

  • Exact special values: Erf[0] = 0, Erf[Infinity] = 1,

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Erf[0]
Out[1]= 0

In[1]:= Erf[Infinity]
Out[1]= 1

In[2]:= Erf[-z]
Out[2]= -Erf[z]

In[1]:= N[Erf[1], 40]
Out[1]= 0.84270079294971486934122063508260925929605

In[1]:= N[Erf[1 + I], 20]
Out[1]= 1.31615128169794764488 + 0.190453469237834686284*I

In[1]:= Series[Erf[x], {x, 0, 7}]
Out[1]= 2/Sqrt[Pi] x + -2/3/Sqrt[Pi] x^3 + 1/5/Sqrt[Pi] x^5 + -1/21/Sqrt[Pi] x^7 + O[x]^8

In[2]:= D[Erf[x^2], x]
Out[2]= (4 x E^(-x^4))/Sqrt[Pi]

Notes

Erf[z] is the error function (2/Sqrt[Pi]) Integral_0^z e^(-t^2) dt, an entire odd function with the exact values Erf[0] = 0 and Erf[±Infinity] = ±1. Real and complex arguments evaluate numerically at machine or arbitrary (MPFR) precision — the complex path uses a DLMF series so N[Erf[1 + I], 20] is correct to the requested digits. The Maclaurin series and the chain-rule derivative D[Erf[z], z] = (2/Sqrt[Pi]) E^(-z^2) are both built in, and Erf is Listable.