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

Real values for real arguments

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.

Simple values at zero and one

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

Simple values at infinity

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

Specific values for specialized arguments

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:

Analyticity

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 .

Poles and essential singularities

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.

Periodicity

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

Parity and symmetry

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:

Series representations

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:

Asymptotic series expansions

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:

Integral representations

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: .

Transformations

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

Representations of derivatives

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:

Differential equations

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: