An Integral on SO(3): Problem 84-1Author(s): Donald RichardsSource: SIAM Review, Vol. 27, No. 1 (Mar., 1985), pp. 81-82Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2031490 .

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PROBLEMS AND SOLUTIONS 81

Editorial note. W. DAHMEN (Universitat Bielefeld, Bielefeld, West Germany) com- ments that this problem is solved in greater generality in his joint paper with C. A. MICCHELI, On entire functions of affine lineage, Proc. Amer. Math. Soc., 84 (1982), pp. 344-346. The paper arose from questions relating to multivariate splines. It is also related to the work of Motzkin and Schoenberg (which is referred to).

An Integral on SO(3)

Problem 84-1*, by B. E. EICHINGER (University of Washington). The following problem is encountered in the classical statistical mechanics of

flexible bodies. As usual, let SO(n) denote the special orthogonal group consisting of n x n orthogonal matrices with determinant + 1. Let a and b be diagonal matrices,

a= diag(al, a 21,an), - oo < al <a2< ... <an< ?')

b=diag(bl,b2 bn), O<b1<b2< bn< oo

and let F (a, b) be the integral

Fn(a,b)=f exp[iTr(ahbh )Idh, SO(n)

where dh is the left-invariant measure on SO(n). For n =2, the evaluation of Fn(a, b) reduces to

F2(a,b) f| exp[iTr{ ah(9)bh'(9)}] d@,

where

()[sin( 8) cos( @)]

Thus one easily finds F2(a, b) = 2gJo(AaAb/2) exp[iTr(a)Tr(b)/2] where Aa = a2 - aI and Ab=b2-bl.

Evaluate F3(a, b). A nice account of integration on SO(3) is given in chapter 4 of Fourier Series and integrals by H. Dym and H. P. McKean (Academic Press, New York, 1972).

Solution by DONALD RICHARDS (University of Wyoming and University of North Carolina). We shall compute the integral

Fn (a,b)=f exp[iTr(ahbhl)Idhs, S0(n)

where SO(n) is the special orthogonal group, a = diag(al,a2,... ,an), b= diag(bl,b2. , bn) with ai, b e R, and dhs is the normalized Haar measure on SO(n). Actually, if 0(n) is the full orthogonal group and dho is the corresponding normalized Haar measure, the function

Gn(a,b)=J exp[iTr(ahbh-1)] dho 0(n)

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82 PROBLEMS AND SOLUTIONS

is important for multivariate statistical theory. It follows from the work of A. T. James and others (cf. Muirhead [1]) that

G ~~~00 ikC(aCKb Gn ( a, b ) = E E C k= k!CK(Ifl)

Here In is the n x n identity matrix; K= (k 1,* * , kn) is a partition of k i.e. k1 ? k2 > *- > k_>0 are nonnegative integers, and k1 + k = k. Also, CK(-) is the zonal poly- nomial corresponding to K; when n =2, CK(*) is expressible in Legendre polynomials. We prove that if n is odd then Fn(a,b)=Gn(a,b). To this end, write 0(n) as the disjoint union of SO(n) and a coset C=h,SO(n), where det(hl)= -1. Then

Gn(a,b)=(f +|) exp[iTr(ahbhl)] dho SO(n)C

F 2 (a, b) +Hj(a, b),

where we have used the relation dhol0SO(n)= 2 dhS, and H2(a,b) denotes the integral over C. To evaluate Hj(a,b), replace h by -h. This maps C bijectively onto SO(n), leaves dh 0 invariant and shows that Hn(a, b) = 2 Fn(a, b).

When n is even, Hn(a, b) is evaluated by replacing h by ph, where p is a permuta- tion matrix chosen to interchange the first two rows of h. It follows from our tech- niques that in this case

Gn(a,b)l = 1 [Fn(a,b) +Fn(ep-rap,b)] .

The problem of evaluating Fn (a, b) was earlier posed by I. Satake [2].

REFERENCES

[1] R. J. MUIRHEAD, Aspects of Multivariate Statistical Theory, John Wiley, New York, 1983. [2] I. SATAKE, Problem 2 in the Problem List from the Special Session on Analysis on Symmetric Complex

Domains, Notices of the AMS, 27 (1980), p. 442.

Zeros of a Definite Integral

Problem 84-2*, by TIAN JINGHUANG (Chengdu Branch, Academia Sinica, Chengdu, China). Prove that the function

F(x, k) = s(x 1) sint k arccos(x-cosX) } d X

has exactly k -1 simple zeros in the interval (0, 2) for k = 1, 2, 3, * - - .

Editorial note. Computations facilitated by writing F as a series of hypergeometric functions lend credence to the proposition, at least for small k. The problem remains open. [0. G. R.]

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