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)
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(2)
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(3)
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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)
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(5)
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(6)
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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)
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(8)
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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)
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can be solved analytically by setting
(10)
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(11)
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(12)
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Then
(13)
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(14)
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For , this is just the usual Gaussian integral, so
(15)
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For , the integrand is integrable by quadrature,
(16)
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To compute for , use the identity
(17)
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(18)
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(19)
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(20)
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For even,
(21)
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(22)
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(23)
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(24)
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(25)
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so
(26)
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(27)
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where is a double factorial. If is odd, then
(28)
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(29)
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(30)
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(31)
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(32)
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so
(33)
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The solution is therefore
(34)
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The first few values are therefore
(35)
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(36)
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(37)
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(38)
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(39)
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(40)
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(41)
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A related, often useful integral is
(42)
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which is simply given by
(43)
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The more general integral of has the following closed forms,
(44)
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(45)
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(46)
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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.