Watson (1939) considered the following three triple integrals,
(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(OEIS A091670, A091671, and A091672), where is a complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function. Analytic computation of these integrals is rather challenging, especially and .
Watson (1939) treats all three integrals by making the transformations
(15)
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(16)
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(17)
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regarding , , and as Cartesian coordinates, and changing to polar coordinates,
(18)
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(19)
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(20)
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after writing .
Performing this transformation on gives
(21)
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(22)
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(23)
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(24)
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can then be directly integrated using computer algebra, although Watson (1939) used the additional transformation
(25)
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to separate the integral into
(26)
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(27)
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(28)
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The integral can also be done by performing one of the integrations
(29)
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with to obtain
(30)
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Expanding using a binomial series
(31)
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(32)
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where is a Pochhammer symbol and
(33)
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Integrating gives
(34)
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(35)
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(36)
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(37)
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Now, as a result of the amazing identity for the complete elliptic integral of the first kind
(38)
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where is the complementary modulus and (Watson 1908, Watson 1939), it follows immediately that with (i.e., , the first singular value),
(39)
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so
(40)
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(41)
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can be transformed using the same prescription to give
(42)
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(43)
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(44)
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(45)
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(46)
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(47)
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where the substitution has been made in the last step. Computer algebra can return this integral in the form of a Meijer G-function
(48)
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but more clever treatment is needed to obtain it in a nicer form. For example, Watson (1939) notes that
(49)
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immediately gives
(50)
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However, quadrature of this integral requires very clever use of a complicated series identity for to obtain term by term integration that can then be recombined as recognized as
(51)
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(Watson 1939).
For , only a single integration can be done analytically, namely
(52)
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It can be reduced to a single infinite sum by defining and using a binomial series expansion to write
(53)
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But this can then be written as a multinomial series and plugged back in to obtain
(54)
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Exchanging the order of integration and summation allows the integrals to be done, leading to
(55)
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Rather surprisingly, the sums over can be done in closed form, yielding
(56)
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where is a generalized hypergeometric function. However, this sum cannot be done in closed form.
Watson (1939) transformed the integral to
(57)
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However, to obtain an entirely closed form, it is necessary to do perform some analytic wizardry (see Watson 1939 for details). The fact that a closed form exists at all for this integral is therefore rather amazing.