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本文深入探讨了误差函数Erf的定义、性质及其在数学分析中的应用。Erf作为正态分布积分的一种规范化形式，是一种在整个复平面上都有定义的全纯函数。文章详细介绍了Erf的级数展开、渐近级数、积分表示以及它与其他特殊函数的关系。

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=============
float erf( float x )
{
   float ax = abs(x);
    float d = 1.0 + 0.278393*ax + 0.230389*ax*ax + 0.000972*ax*ax*ax + 0.078108*ax*ax*ax*ax;
    return sign(x)*(1.0 - 1.0/(d*d*d*d));

return 1 - 2 / ( 1 + exp2( 3.35 * x ) );
//return 1 - 2 / ( 1 + exp2( 3.35 * x ) ) + 0.07219 * x * (15.41819 * Square(x*x) - 1) * exp2( -5.609846 * abs(x) );
}

G:\File\opensource\UnrealEngine-4.0\UnrealEngine-git\Engine\Shaders\Private\RectLight.ush

===========
Erf
http://mathworld.wolfram.com/Erf.html

Calculus and Analysis > Special Functions > Erf >

Calculus and Analysis > Complex Analysis > Entire Functions >

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Erf

is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by

(1)

Note that some authors (e.g., Whittaker and Watson 1990, p. 341) define without the leading factor of .

Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as Erf[z0, z1].

Erf satisfies the identities

			(2)
			(3)
			(4)

where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For ,

(5)

where is the incomplete gamma function.

Erf can also be defined as a Maclaurin series

			(6)
			(7)

(OEIS A007680). Similarly,

(8)

(OEIS A103979 and A103980).

For , may be computed from

			(9)
			(10)

(OEIS A000079 and A001147; Acton 1990).

For ,

			(11)
			(12)

Using integration by parts gives

			(13)
			(14)
			(15)
			(16)

(17)

and continuing the procedure gives the asymptotic series

			(18)
			(19)
			(20)

(OEIS A001147 and A000079).

Erf has the values

			(21)
			(22)

It is an odd function

(23)

and satisfies

(24)

Erf may be expressed in terms of a confluent hypergeometric function of the first kind as

			(25)
			(26)

Its derivative is

(27)

where is a Hermite polynomial. The first derivative is

(28)

and the integral is

(29)

	Min	Max
Re
Im

Erf can also be extended to the complex plane, as illustrated above.

A simple integral involving erf that Wolfram Language cannot do is given by

(30)

(M. R. D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include

int_0^infty(e^(-(p+x)y))/(pi(p+x))sin(asqrt(x))dx=-sinh(asqrt(p)) 
 +(e^(-asqrt(p)))/2erf(a/(2sqrt(y))-sqrt(py))+(e^(asqrt(p)))/2erf(a/(2sqrt(y))+sqrt(py)) 
int_0^infty(sqrt(x)e^(-(p+x)y))/(pi(p+x))cos(asqrt(x))dx=(e^(-[py+a^2/(4y)]))/(sqrt(piy))+sqrt(p)[-cosh(asqrt(p))-(e^(-asqrt(p)))/2erf(a/(2sqrt(y))-sqrt(py))+(e^(asqrt(p)))/2erf(a/(2sqrt(y))+sqrt(py))]

(31)

(M. R. D'Orsogna, pers. comm., Dec. 15, 2005).

Erf has the continued fraction

			(32)
			(33)

(Wall 1948, p. 357), first stated by Laplace in 1805 and Legendre in 1826 (Olds 1963, p. 139), proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp. 8-9).

Definite integrals involving include Definite integrals involving include

			(34)
			(35)
			(36)
			(37)
			(38)

The first two of these appear in Prudnikov et al. (1990, p. 123, eqns. 2.8.19.8 and 2.8.19.11), with , .

A complex generalization of is defined as

			(39)
			(40)

Integral representations valid only in the upper half-plane are given by

			(41)
			(42)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Error Function and Fresnel Integrals." Ch. 7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 297-309, 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.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 105, 2003.

Olds, C. D. Continued Fractions. New York: Random House, 1963.

Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 2: Special Functions. New York: Gordon and Breach, 1990.

Sloane, N. J. A. Sequences A000079/M1129, A001147/M3002, A007680/M2861, A103979, A103980 in "The On-Line Encyclopedia of Integer Sequences."

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.

Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.

Watson, G. N. "Theorems Stated by Ramanujan (IV): Theorems on Approximate Integration and Summation of Series." J. London Math. Soc. 3, 282-289, 1928.

Whittaker, E. T. and Robinson, G. "The Error Function." §92 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 179-182, 1967.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Referenced on Wolfram|Alpha: Erf CITE THIS AS:

Weisstein, Eric W. "Erf." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Erf.html