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
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
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
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
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(9)
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can be solved analytically by setting
Then
For , this is just the usual Gaussian integral,
so
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(15)
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For , the integrand is integrable by quadrature,
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(16)
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To compute for , use the
identity
For even,
so
where is a double factorial. If is odd, then
so
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(33)
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The solution is therefore
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(34)
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The first few values are therefore
A related, often useful integral is
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(42)
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which is simply given by
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(43)
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The more general integral of
has the following closed forms,
for integer (F. Pilolli, pers. comm.).
For (45) and (46),
(the punctured plane), , and . Here, is a confluent
hypergeometric function of the second kind and is a binomial coefficient.
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.
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