The bestknown 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, closedform 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 secondorder linear differential equations:
where and are arbitrary constants.
The inverses of the probability integrals , , and satisfy the following ordinary secondorder nonlinear differential equations:
