570 



BELL SYSTEM TECHNICAL JOURNAL 



Bessel Function 

 For the Bessel integral with h large compared with n we have 



;^ = 0, u= \, g(0) - 1/2, F- 1, 



</)(x) 



u 



1 — Sin^ I ;r 



= lixgY - {xgyy 



r=0 \ ' 



Substituting in (8) we have 



a/_ IV' ^ ( n \ [P/-2«x'^»+2rg-(«-2n-2.-2r+l)-[^^p 



5! '-hv\h^ ^ \rl (5-2.)! 



where 2R> s - 2n - 2v] note that li R > n, then ( 1=0. 



Since ^s = for 5 < 2n and g is an even function of x, set 5 = 2w. 

 + 2w. Then Leibnitz's theorem and interchanging v with m — v give 



(11) 



^2n 



+2m 



m „.,m—v 



= L 



w" 



(2w + 2m) ! „T'o (w? — ^) 



l.t-o^ ^ U/ (2z;-2r)! 



where again we may have recourse to (10) for the differentiations of 

 negative powers of g{x). 



Appendix II 

 Writing the incomplete Beta Function in the form 



I(P) 



= ^(1 - xy 



Jo 



+w-^c-i(^l _ x)~'-''+"''>dx, 



we now have 



N = n -\- lu, Xi = 0, X2 = p, 

 y{x) = (1 - x), F= 1, X= 0, 



(l)n,{x) = x'-'^{l - x)-'-'+"'\ 

 log F - log y = - log (1 - x), 



(X x^ \ 



1 + 2 "^ V " * ) ' 



u = 0, 

 y(x) = e~\ 

 T,= 0, 



n = - (w + ^to log (1 - ^). 



I 



