AN EXPANSION FOR LAPLACIAN INTEGRALS 567 



for b large compared with w, n being a positive integer. Writing 



e-'LXb) = -7-: 



^| Trr(n + 



leads us to the modified Laplacian expansion 



2 b 



2) Jo 



<TT{2b + W) 



where 



" .feo\86 + 4w) 2 (2«) ! P{n + i 2& + 7t;) (w + m) ! 



To determine the coefficients A^n+zm we proceed as follows: 

 For the integral now under consideration 



N = 2b -\- W, Xi = 0, X2 = TT, 



y(^x) = e-^^-^^"^''), X = 0, F = 1, 

 (log F - log 3-)^ = (i - I cosx)^ = xg{x), ■ 



\ . X ^ (- lyx^" 



g{x) = - sm - = 2. 



x 2 fcro22^+H2^ + 1)!' 

 so that 



u = 1, 3;(x) = e-'\ T, = 0, Ts = 2& + w, 



0t.(:v;) = (I sinx)2»g"'C-<')% 



Bin+2m = InWin + |)/(w + w)!(2»)!22'"+^]^2n+2,n. 



The relations above give 



4>^{x){dx/dt) = 2/2" (1 - /2)n-Jg«,<2, 



Let 



E^'2«+2./2"+2V(2w + 25)! = 2/2"(l - /2)«-^, 

 s=0 



so that 



A',n- 2(2w)!, 



_ (- 1)^(2/^ - l){2n - 3) ••' {2n - 25 + 1)(2^^ + 25)! , 



^ 2n+2s — \ 2«-l5l "^ ~ ' 



then 



= Z Z^'"+'^+''^'2n+2.wV^!(2w + 25)! 

 A.-=0 x=0 



3 An expansion of this type for Bessel functions was derived many years ago by 

 Hadamard; see Watson, "A Treatise on the Theory of Bessel Functions," pp. 2U4 

 and 205. 



