Boersma's solution to Anicin's problem

[thk (Tom H. Koornwinder)]

>From boersma@win.tue.nl Thu Jan 22 13:06 MET 1998
>To: anicin@EUnet.yu
>Subject: Problem in OP-SF NET 5.1
 
   Dear Dr. Anicin,
 
Here is a solution of your problem as published by Koornwinder
in the OP-SF NET 5.1, dated January 15, 1998.
 
Clearly, your function g(z) is related to the Fourier cosine 
transform of  I_1(a) K_1(a). This transform is listed in 
[ 1, form. 1.12(47)]. By inversion of the latter transform one has
 
2 I_1(a) K_1(a) = (4/pi) \int_0^\infty Q_(1/2)(1+2z^2)cos(2az) dz
 
and, for a = 0,
 
1 = (4/pi) \int_0^\infty Q_(1/2)(1+2z^2) dz.
 
Next, the Legendre function  Q_(1/2)  (that is, Q with subscript 1/2) 
is expressible in terms of elliptic integrals K(k) and E(k) :
 
Q_(1/2)(1+2z^2) = (2/k - k) K(k) - (2/k) E(k)
 
where  k = (1+z^2)^(-1/2); see e.g. [ 2, form. 8.13.7]. 
I believe this completely solves your problem.
 
References
[1] A. Erdelyi, W. Magnus, F. Oberhettinger, & F.G. Tricomi, Tables 
     of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1954.
[2] M. Abramowitz & I.A. Stegun, Handbook of Mathematical Functions,
     Dover Publications, New York, 1965.
 
 
Sincerely yours,
John Boersma