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Gaussian Integral

المؤلف:  Guitton, E.

المصدر:  "Démonstration de la formule." Nouv. Ann. Math. 65

الجزء والصفحة:  ...

28-4-2019

4250

Gaussian Integral

 

The Gaussian integral, also called the probability integral and closely related to the erf function, is the integral of the one-dimensional Gaussian function over . It can be computed using the trick of combining two one-dimensional Gaussians

			(1)
			(2)
			(3)

Here, use has been made of the fact that the variable in the integral is a dummy variable that is integrates out in the end and hence can be renamed from  to . Switching to polar coordinates then gives

			(4)
			(5)
			(6)

There also exists a simple proof of this identity that does not require transformation to polar coordinates (Nicholas and Yates 1950).

The integral from 0 to a finite upper limit  can be given by the continued fraction

			(7)
			(8)

where  is erf (the error function), as first stated by Laplace, proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp. 8-9).

The general class of integrals of the form

(9)

can be solved analytically by setting

			(10)
			(11)
			(12)

Then

			(13)
			(14)

For , this is just the usual Gaussian integral, so

(15)

For , the integrand is integrable by quadrature,

(16)

To compute  for , use the identity

			(17)
			(18)
			(19)
			(20)

For  even,

			(21)
			(22)
			(23)
			(24)
			(25)

so

			(26)
			(27)

where  is a double factorial. If  is odd, then

			(28)
			(29)
			(30)
			(31)
			(32)

so

(33)

The solution is therefore

{((n-1)!!)/(2^(n/2+1)a^(n/2))sqrt(pi/a) for n even; ([1/2(n-1)]!)/(2a^((n+1)/2)) for n odd. " src="http://mathworld.wolfram.com/images/equations/GaussianIntegral/NumberedEquation5.gif" style="height:98px; width:286px" />

(34)

The first few values are therefore

			(35)
			(36)
			(37)
			(38)
			(39)
			(40)
			(41)

A related, often useful integral is

(42)

which is simply given by

{(2I_n(a))/(sqrt(pi)) for n even; 0 for n odd. " src="http://mathworld.wolfram.com/images/equations/GaussianIntegral/NumberedEquation7.gif" style="height:64px; width:178px" />

(43)

The more general integral of  has the following closed forms,

			(44)
			(45)
			(46)

for integer  (F. Pilolli, pers. comm.). For (45) and (46), {0}" src="http://mathworld.wolfram.com/images/equations/GaussianIntegral/Inline132.gif" style="height:14px; width:78px" /> (the punctured plane), , and . Here,  is a confluent hypergeometric function of the second kind and  is a binomial coefficient.

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REFERENCES:

Guitton, E. "Démonstration de la formule." Nouv. Ann. Math. 65, 237-239, 1906.

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

Nicholas, C. B. and Yates, R. C. "The Probability Integral." Amer. Math. Monthly 57, 412-413, 1950.

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 147-148, 1984.

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