Watson's Triple Integrals

Watson (1939) considered the following three triple integrals,


(OEIS A091670, A091671, and A091672), where K(k) is a complete elliptic integral of the first kind, theta_3(0,q) is a Jacobi theta function, and Gamma(z) is the gamma function. Analytic computation of these integrals is rather challenging, especially I_2 and I_3.

Watson (1939) treats all three integrals by making the transformations


regarding x, y, and z as Cartesian coordinates, and changing to polar coordinates,


after writing 2phi=psi.

Performing this transformation on I_1 gives


I_1 can then be directly integrated using computer algebra, although Watson (1939) used the additional transformation


to separate the integral into


The integral I_1 can also be done by performing one of the integrations


with c=cosvcosw to obtain


Expanding using a binomial series


where (z)_n is a Pochhammer symbol and


Integrating gives


Now, as a result of the amazing identity for the complete elliptic integral of the first kind K(k)


where k^' is the complementary modulus and 0<k<=1/sqrt(2) (Watson 1908, Watson 1939), it follows immediately that with k=k^'=1/sqrt(2) (i.e., k=k_1, the first singular value),




I_2 can be transformed using the same prescription to give


where the substitution t=tantheta has been made in the last step. Computer algebra can return this integral in the form of a Meijer G-function

 I_2=1/(2pi^(5/2))G_(3,3)^(3,2)(4|1/2,1/2,1/2; 0,0,0),

but more clever treatment is needed to obtain it in a nicer form. For example, Watson (1939) notes that


immediately gives


However, quadrature of this integral requires very clever use of a complicated series identity for K(k) to obtain term by term integration that can then be recombined as recognized as


(Watson 1939).

For I_3, only a single integration can be done analytically, namely


It can be reduced to a single infinite sum by defining w=(cosx+cosy+cosz)/3 and using a binomial series expansion to write


But this can then be written as a multinomial series and plugged back in to obtain

 I_3=1/(pi^3)int_(-pi)^piint_(-pi)^piint_(-pi)^pisum_(k=0)^infty1/(3^(k+1))×sum_(n_1,n_2,n_3>=0; n_1+n_2+n_3=k)(k!)/(n_1!n_2!n_3!)cos^(n_1)xcos^(n_2)ycos^(n_3)zdxdydz.

Exchanging the order of integration and summation allows the integrals to be done, leading to

 I_3=(pi^3)/3sum_(k=0)^infty1/(3^k)sum_(n_1,n_2,n_3>=0; n_1+n_2+n_3=k)(k!)/(n_1!n_2!n_3!) 

Rather surprisingly, the sums over n_i can be done in closed form, yielding


where _3F_2(a,b,c;d,e;z) is a generalized hypergeometric function. However, this sum cannot be done in closed form.

Watson (1939) transformed the integral to


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.

See also

Pólya's Random Walk Constants, Watson's Formula, Watson's Identities

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Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006b.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Domb, C. "On Multiple Returns in the Random-Walk Problem." Proc. Cambridge Philos. Soc. 50, 586-591, 1954.Glasser, M. L. and Zucker, I. J. "Extended Watson Integrals for the Cubic Lattices." Proc. Nat. Acad. Sci. U.S.A. 74, 1800-1801, 1977.Joyce, G. and Zucker, I. J. "On the Evaluation of Generalized Watson Integrals." Proc. Amer. Math. Soc. 133, 71-81, 2005.McCrea, W. H. and Whipple, F. J. W. "Random Paths in Two and Three Dimensions." Proc. Roy. Soc. Edinburgh 60, 281-298, 1940.Sloane, N. J. A. Sequences A091670, A091671, and A091672 in "The On-Line Encyclopedia of Integer Sequences."Watson G. N. "The Expansion of Products of Hypergeometric Functions." Quart. J. Pure Appl. Math. 39, 27-51, 1907.Watson G. N. "A Series for the Square of the Hypergeometric Function." Quart. J. Pure Appl. Math. 40, 46-57, 1908.Watson, G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 2 10, 266-276, 1939.

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Watson's Triple Integrals

Cite this as:

Weisstein, Eric W. "Watson's Triple Integrals." From MathWorld--A Wolfram Web Resource.

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