Solution to problem 75-20 : Limit of an integral

Solution to problem 75-20 : Limit of an integral

Boersma, J. (1976). Solution to problem 75-20 : Limit of an integral. SIAM Review, 18(4), 770-772.

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770 MURRAY S. KLAMKIN

This divergent series was summed by Euler, as shown in Hardy [1, p. 26], to yield

F(t) = _C

g

1+ Ctw as the continuous solution of the problem posed.

REFERENCE

[1] G. H. HARDY, Divergent Series, Oxford University Press, London, 1949.

Also solved by 0. P. LosSERS (Technological University, Eindhoven, the Netherlands) and the proposer, both of whom verified the solution by substitu- tion.

Comment by the proposer. The calculation of the coefficients is closely related to the following sum identity

n

₁

due to TOR B. STOVER (see Gould, Combinational Identities, Morgantown, W. Va., 1972).

Problem 75-20, Limit of an Integral, by M. L. GLASSER (University of Water- loo, Ontario, Canada).

Show that

lim n IIn (x) Jn (x)Kn (x) dx = 8-112

n-*ooo

where, as usual, I,,, J,,, Kn are Bessel functions.

Solution by J. BOERSMA (Technical University, Eindhoven, the Nether- lands).

Let the integral be denoted by A,,. Then it is shown in two different ways that

lim,,oo An = 2 .

Firstly, replace the integration variable x by nx, yielding

00

(1) An = n2

J

In the latter integral we insert the uniform asymptotic expansion

(2) In (nx)Kn (nx)= - 1 [1 +O(n')], n- oo, x'0,

PROBLEMS AND SOLUTIONS 771

x ? 0. Then the leading term of An becomes, by means of [2, formula 8.5 (11)] and (2),

00

(3) ₂ (1 _{+x2)n12J(nx) dx = nIn12(4n)Knl2(-n)} = 2 3/2[1 + (n

The contribution of the remainder term in (2) can be estimated by means of the inequality (cf. Watson [3, formulas 8.5(9), 13.74(1)])

(4)

IJn

(nx)I

7Tn

thus yielding

00

(5)

j

Then it is found from (3) and (5) that

(6) An= 2-3/2 + o(n-1/2) lim An - 2-3/2.

n-+o0

Secondly, by means of [2, formula 8.13(20)], An can be expressed in terms of Legendre functions, viz.,

(7) An= e F( n + 1) P /2 1/2(5)Q n/2-1/2(5)

Using Whipple's formula (cf. [4, formulas 3.3.1(13), (14)]), we can reduce the latter result to

nF(3n

+41)

(8) A _{e nii-i/2} 2 2 _{Pn/2112(14)Qfn 2(2J)}

2F(ln4 n 1/ -1 n-/2 2

When n -- _{oo, one needs asymptotic expansions of Legendre functions of large}

degree and order. Such asymptotic expansions were derived by Thorne [5]. When specializing Thorne's results to the present case, one gets

F(l n

+4

(9) _{Pn-1/2(42)n-Q/2(241)} = 2-l/2e1n5i/2 n(3 +n[l f+(- ]

n -o 0.

Consequently,

(10) An = 2 -3/2[ 1 - 15n -2 + 0(n -4)], n o x,

and limn,oO An = 2 3/2, as found before.

The asymptotic expansion (10) has been checked by the first approach using a proper extension of Olver's uniform asymptotic expansion (2).

772 MURRAY S. KLAMKIN

Finally, Mr. J. K. M. JANSEN of our University determined the numerical value of An for n = 5, 10, 20 from (7):

A5 = 0.350412, A10= 0.352736, A20,-0.353347,

correct to six decimal places. These values do agree very well with (10); e.g., for n = 10 the error turns out to be 10 5. By extrapolation from A5, A1o, A20, Jansen

found the "numerical" limit to be 0.353553, which is identical to 2 3/2 in all six

decimal places.

REFERENCES

[1] F. W. J. OLVER, Asymptotics and Special Functions, Academic Press, New York, 1974.

[2] A. ERDELYI, W. MAGNUS, F. OBERHETFINGER AND F. G. TRICOMI, Tables of Integral

Transforms, vol. II, McGraw-Hill, New York, 1954.

[3] G. N. WATSON, A Treatise on the Theory of Bessel Functions, 2nd Ed., Cambridge University Press, Cambridge, 1958.

[4] A. ERDIELYI, W. MAGNUS, F. OBERHETrINGER AND F. G. TRICOMI, Higher Transcendental

Functions, vol. I, McGraw-Hill, New York, 1953.

[5] R. C. THORNE, The asymptotic expansion of Legendre functions of large degree and order, Philos. Trans. Roy. Soc. London Ser. A, 249(1957), pp. 597-620.

Also solved by D. E. AMOS (Sandia Laboratories) and A. G. GIBBS (Battelle

Memorial Institute), who both showed that

00

lim n

In

Editorial note. Procedures for obtaining the asymptotic behavior of the latter integral and more general ones are to appear in a paper by F. W. J. OLVER and the

proposer.

Problem 75-21, n-dimensional Simple Harmonic Motion, by I. J. SCHOENBERG

(University of Wisconsin). In Rn, we consider the curve

(1) F: xi =cos (Ait+ai), i=1, ,n, -xC<tC<x.

which represents an n-dimensional simple harmonic motion entirely contained within the cube U: -1?xi?1, i= 1, , n. We want F to be truly n- dimensional and will therefore assume without loss of generality that

(2) Ai > 0 for all i.

We consider the open sphere

n

S: E 2 2 i=l

and want the motion (1) to take place entirely outside of S, hence contained in the closed set U- S. What is the largest sphere S such that there exist motions F