“Erf”的概念、定义、习题解题方法及翻译-数学辞典

单词

Erf

释义

Erf

The ``error function'' encountered in integrating the Gaussian Distribution.

			(1)
			(2)
			(3)

where Erfc is the complementary error function and

is the incomplete Gamma Function. It canalso be defined as a Maclaurin Series

(4)

Erf has the values

			(5)
			(6)

It is an Odd Function

(7)

and satisfies

(8)

Erf may be expressed in terms of a Confluent Hypergeometric Function of the First Kind

(9)

Erf is bounded by

(10)

Its Derivative is

(11)

where

is a Hermite Polynomial. The first Derivative is

(12)

and the integral is

(13)

For , erf may be computed from

			(14)


			(15)

			(16)
			(17)

(Acton 1990). For


			(18)

Using Integration by Parts gives




			(19)

(20)

and continuing the procedure gives the Asymptotic Series

(21)

A Complex generalization of is defined as

			(22)
			(23)
			(24)

See also Dawson's Integral,Erfc, Erfi, Gaussian Integral, Normal Distribution Function, Probability Integral

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Error Function'' and ``Repeated Integrals of the Error Function.'' §7.1-7.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 297-300, 1972.

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 16, 1990.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 568-569, 1985.

Spanier, J. and Oldham, K. B. ``The Error Function and Its Complement .'' Ch. 40 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 385-393, 1987.

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