Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Erf






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erf[z1,z2] > Introduction to the probability integrals and inverses





The best-known properties and formulas for probability integrals and inverses


For real values of argument , the values of the probability integrals , , , and are real. For real arguments , the values of the inverse error function are real; for real arguments , the values of the inverse of the generalized error function are real; and for real arguments , the values of the inverse complementary error function are real.

The probability integrals , , , and , and their inverses , , and have simple values for zero or unit arguments:

The probability integrals , , and have simple values at infinity:

In cases when or is equal to or , the generalized error function and its inverse can be expressed through the probability integrals , , or their inverses by the following formulas:

The probability integrals , , and , and their inverses , and are defined for all complex values of , and they are analytical functions of over the whole complex ‐plane. The probability integrals , , and are entire functions with an essential singular point at , and they do not have branch cuts or branch points.

The generalized error function is an analytical function of and , which is defined in . For fixed , it is an entire function of . For fixed , it is an entire function of . It does not have branch cuts or branch points. The inverse of the generalized error function is an analytical function of and , which is defined in .

The probability integrals , , and have only one singular point at . It is an essential singular point.

The generalized error function has singular points at and . They are essential singular points.

The probability integrals , , , and , and their inverses , , and do not have periodicity.

The probability integrals , , and are odd functions and have mirror symmetry:

The generalized error function has permutation symmetry:

The complementary error function has mirror symmetry:

The probability integrals , , , and , and their inverses and have the following series expansions:

The series for functions , , , and converge for all complex values of their arguments.

Interestingly, closed-form expressions for the truncated version of the Taylor series at the origin can be expressed through generalized hypergeometric function , for example:

The asymptotic behavior of the probability integrals , , and can be described by the following formulas (only the main terms of the asymptotic expansion are given):

The previous formulas are valid in any direction approaching infinity (z∞). In particular cases, these formulas can be simplified to the following relations:

The probability integrals , , , and can also be represented through the following equivalent integrals:

The symbol in the preceding integral means that the integral evaluates as the Cauchy principal value: .

If the arguments of the probability integrals , , and contain square roots, the arguments can sometimes be simplified:

The derivative of the probability integrals , , , and , and their inverses , , and have simple representations through elementary functions:

The symbolic -order derivatives from the probability integrals , , , and have the following simple representations through the regularized generalized hypergeometric function :

But the symbolic -order derivatives from the inverse probability integrals , , and have very complicated structures in which the regularized generalized hypergeometric function appears in the multidimensional sums, for example:

The probability integrals , , , and satisfy the following second-order linear differential equations:

where and are arbitrary constants.

The inverses of the probability integrals , , and satisfy the following ordinary second-order nonlinear differential equations: