Watson's Triple Integrals

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Watson (1939) considered the following three triple integrals,

👁 I_1	👁 =	👁 1/(pi^3)int_0^piint_0^piint_0^pi(dudvdw)/(1-cosucosvcosw)	(1)
👁 Image	👁 =	👁 (4[K(1/2sqrt(2))]^2)/(pi^2)	(2)
👁 Image	👁 =	👁 (Gamma^4(1/4))/(4pi^3)	(3)
👁 Image	👁 =	👁 1.39320393...	(4)
👁 I_2	👁 =	👁 1/(pi^3)int_0^piint_0^piint_0^pi(dudvdw)/(3-cosvcosw-coswcosu-cosucosv)	(5)
👁 Image	👁 =	👁 (sqrt(3)[K(1/4(sqrt(6)-sqrt(2)))]^2)/(pi^2)	(6)
👁 Image	👁 =	👁 (3Gamma^6(1/3))/(2^(14/3)pi^4)	(7)
👁 Image	👁 =	👁 0.448220394...	(8)
👁 I_3	👁 =	👁 1/(pi^3)int_0^piint_0^piint_0^pi(dudvdw)/(3-cosu-cosv-cosw)	(9)
👁 Image	👁 =	👁 (4(18+12sqrt(2)-10sqrt(3)-7sqrt(6))[K((2-sqrt(3))(sqrt(3)-sqrt(2)))]^2)/(pi^2)	(10)
👁 Image	👁 =	👁 (18+12sqrt(2)-10sqrt(3)-7sqrt(6))[1+2sum_(k=1)^(infty)exp(-k^2pisqrt(6))]^4	(11)
👁 Image	👁 =	👁 (18+12sqrt(2)-10sqrt(3)-7sqrt(6))theta_3^4(0,e^(-pisqrt(6)))	(12)
👁 Image	👁 =	👁 (sqrt(6))/(96pi^3)Gamma(1/(24))Gamma(5/(24))Gamma(7/(24))Gamma((11)/(24))	(13)
👁 Image	👁 =	👁 0.505462019...	(14)

(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

👁 x	👁 =	👁 tan(1/2u)	(15)
👁 y	👁 =	👁 tan(1/2v)	(16)
👁 z	👁 =	👁 tan(1/2w),	(17)

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

👁 x	👁 =	👁 rsinthetacosphi	(18)
👁 y	👁 =	👁 rsinthetasinphi	(19)
👁 z	👁 =	👁 rcostheta	(20)

after writing 👁 2phi=psi
.

Performing this transformation on 👁 I_1
gives

👁 I_1	👁 =	👁 8/(pi^3)int_0^inftyint_0^inftyint_0^infty(dxdydz)/((1+x^2)(1+y^2)(1+z^2)-(1-x^2)(1-y^2)(1-z^2))	(21)
👁 Image	👁 =	👁 4/(pi^3)int_0^inftyint_0^inftyint_0^infty(dxdydz)/(x^2+y^2+z^2+x^2y^2z^2)	(22)
👁 Image	👁 =	👁 4/(pi^3)int_0^(pi/2)int_0^(pi/2)int_0^infty(sinthetadrdthetadphi)/(1+r^4sin^4thetacos^2thetasin^2phicos^2phi)	(23)
👁 Image	👁 =	👁 4/(pi^3)int_0^(pi/2)int_0^(pi/2)int_0^infty(sinthetadrdthetadpsi)/(1+1/4r^4sin^4thetacos^2thetasin^2psi).	(24)

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

👁 t=rsinthetasqrt(1/2costhetasinpsi)

(25)

to separate the integral into

👁 I_1	👁 =	👁 (4sqrt(2))/(pi^3)int_0^infty(dt)/(1+t^4)int_0^(pi/2)(dtheta)/(sqrt(costheta))int_0^(pi/2)(dpsi)/(sqrt(sinpsi))	(26)
👁 Image	👁 =	👁 (4sqrt(2))/(pi^3)·pi/(2sqrt(2))·(Gamma^2(1/4))/(2sqrt(2pi))·(Gamma^2(1/4))/(2sqrt(2pi))	(27)
👁 Image	👁 =	👁 (Gamma^4(1/4))/(4pi^3).	(28)

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

👁 int_0^pi(du)/(1-ccosu)=pi/(sqrt(1-c^2))

(29)

with 👁 c=cosvcosw
to obtain

👁 I_1=1/(pi^2)int_0^piint_0^pi(dvdw)/(sqrt(1-cos^2vcos^2w)).

(30)

Expanding using a binomial series

👁 (1-c)^(-1/2)	👁 =	👁 sum_(n=0)^(infty)((1/2)_n)/(n!)c^n	(31)
👁 Image	👁 =	👁 sum_(n=0)^(infty)a_nc^n	(32)

where 👁 (z)_n
is a Pochhammer symbol and

👁 a_n=((2n-1)!!)/((2n)!!).

(33)

Integrating gives

👁 I_1	👁 =	👁 1/(pi^2)int_0^piint_0^pisum_(n=0)^(infty)a_ncos^(2n)vcos^(2n)wdvdw	(34)
👁 Image	👁 =	👁 1/(pi^2)sum_(n=0)^(infty)a_n(int_0^picos^(2n)vdv)^2	(35)
👁 Image	👁 =	👁 1/(pi^2)sum_(n=0)^(infty)a_n(pia_n)^2	(36)
👁 Image	👁 =	👁 sum_(n=0)^(infty)a_n^3.	(37)

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

👁 [K(k)]^2=1/4pi^2sum_(n=0)^inftya_n^3(2kk^')^(2n),

(38)

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),

👁 K(1/2sqrt(2))=1/4pi^2sum_(n=0)^inftya_n^3=1/4pi^2I_1,

(39)

👁 I_1	👁 =	👁 (4[K(1/2sqrt(2))]^2)/(pi^2)	(40)
👁 Image	👁 =	👁 (Gamma^4(1/4))/(4pi^3).	(41)

👁 I_2
can be transformed using the same prescription to give

👁 I_2	👁 =	👁 2/(pi^3)int_0^inftyint_0^inftyint_0^infty(dxdydz)/(3product(1+x^2)-sum(1-y^2)(1-z^2)(1+x^2))	(42)
👁 Image	👁 =	👁 2/(pi^3)int_0^inftyint_0^inftyint_0^infty(dxdydz)/(x^2+y^2+z^2+y^2z^2+z^2x^2+x^2y^2)	(43)
👁 Image	👁 =	👁 2/(pi^3)int_0^(pi/2)int_0^(pi/2)int_0^infty(sinthetadrdthetadphi)/(1+r^2sin^2theta(cos^2theta+sin^2psisin^2phicos^2phi))	(44)
👁 Image	👁 =	👁 2/(pi^3)int_0^(pi/2)int_0^(pi/2)int_0^infty(sinthetadrdthetadpsi)/(1+r^2sin^2theta(cos^2theta+1/4sin^2thetasin^2psi))	(45)
👁 Image	👁 =	👁 1/(pi^2)int_0^(pi/2)int_0^(pi/2)(dthetadpsi)/(sqrt(cos^2theta+1/4sin^2thetasin^2psi))	(46)
👁 Image	👁 =	👁 1/(pi^2)int_0^(pi/2)int_0^infty(dtdpsi)/(sqrt((1+t^2)(1+1/4t^2sin^2psi))),	(47)

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),

(48)

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

👁 K^'(k^')=int_0^infty(dt)/(sqrt((1+t^2)(1+k^('2)t^2)))

(49)

immediately gives

👁 I_2=1/(pi^2)int_0^piK^'(1/2sinpsi)dpsi.

(50)

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

👁 I_2=(K(k_1)K(k_1^'))/(pi^2)

(51)

(Watson 1939).

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

👁 int_(-pi)^pi1/(3-cosx-cosy-cosz)dz =-(2pi)/(sqrt((cosx+cosy-2)(cosx+cosy-4))).

(52)

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

👁 1/w=1/3sum_(k=0)^inftyw^k=sum_(k=0)^infty1/(3^(k+1))(cosx+cosy+cosz)^k.

(53)

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.

(54)

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!) ×((-1)^(n_1+n_2+n_3)2^(n_1+n_2+n_3))/(Gamma^2(1/2(1-n_1))Gamma^2(1/2(1-n_2))Gamma^2(1/2(1-n_3))).

(55)

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

👁 I_3=1/3sum_(n=0)^infty([(2n)!]^2)/(36^n(n!)^3)_3F_2(-n,-n,-n;1,1/2-n;1/4),

(56)

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

👁 I_3=(2sqrt(2))/piint_0^(pi/2)int_0^infty(dtdpsi)/(sqrt((1+4t^2+3t^4sin^2psi)(1+t^2sin^2psi))).

(57)

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.

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References

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 Resource. https://mathworld.wolfram.com/WatsonsTripleIntegrals.html

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