Inverse Error Function Asymptotic
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(1774) where it was expressed through the following integral: Later C. Kramp (1799) used this integral for the definition of the complementary error function . P.‐S. Laplace erf(1) (1812) derived an asymptotic expansion of the error function. The probability integrals complementary error function table were so named because they are widely applied in the theory of probability, in both normal and limit error function excel distributions. To obtain, say, a normal distributed random variable from a uniformly distributed random variable, the inverse of the error function, namely is needed. The inverse was systematically investigated in http://www.ams.org/mcom/1976-30-136/S0025-5718-1976-0421040-7/S0025-5718-1976-0421040-7.pdf the second half of the twentieth century, especially by J. R. Philip (1960) and A. J. Strecok (1968).
Definitions of probability integrals and inverses The probability integral (error function) , the generalized error function , the complementary error function , the imaginary error function , the inverse error function , the inverse of the generalized error function , and http://functions.wolfram.com/GammaBetaErf/InverseErf/introductions/ProbabilityIntegrals/ShowAll.html the inverse complementary error function are defined through the following formulas: These seven functions are typically called probability integrals and their inverses. Instead of using definite integrals, the three univariate error functions can be defined through the following infinite series. A quick look at the probability integrals and inversesHere is a quick look at the graphics for the probability integrals and inverses along the real axis. Connections within the group of probability integrals and inverses and with other function groups Representations through more general functions The probability integrals , , , and are the particular cases of two more general functions: hypergeometric and Meijer G functions. For example, they can be represented through the confluent hypergeometric functions and : Representations of the probability integrals , , , and through classical Meijer G functions are rather simple: The factor in the last four formulas can be removed by changing the classical Meijer G functions to the generalized one: The probability integrals , , , and are the particular cases of the incomplete gamma function, regularized incomplete gamma futext Help pages Full-text links: Download: http://arxiv.org/abs/math/0607230 PDF PostScript Other formats (license) Current browse context: math