568 



BELL SYSTEM TECHNICAL JOURNAL 



and 



^2n+2n./(2w + 2m) ! = Z (^V-^ 0^'2n+2m-2 J(2« + 2m - 2k)\ 



We must now assign a value to w. One which suggests itself is to 

 take w such that the second term in our expansion for /„(&) vanishes; 

 in other words, such that ^2n+2 shall be zero. This merely requires 

 that we set w = n — Xjl, giving 



A2n = 2{2n)\, A2n+2= 



^2n+4 = - {2n- l)(2w + 4)!/2, ylgn+e = - {2n - l){2n + 6)!/3 

 A2n+s = {2n - l)(2n - 5){2n + 8)!/16, etc. 

 For n = 1 and w = 1/2 we have 



A^ =4, yl4 = 0, ^6 = - 360, ^8 = - 13440, • • • 

 2b 



e-H,{b) = 



2b + .5 



\p(3/2,2b + .5)/Vx(2& + .5) 



5i 



wherein 



5i = 1 - 15(8i + 2)-2 



P(7/2, 2b + .5) 

 P(3/2, 26 + .5) 



- 140(86 + 2)-3 



P(9/2, 26 + .5) 

 P(3/2, 26 + .5) 



The degree of accuracy obtainable when only one, two or three 

 terms of Si are made use of is indicated by the second table at the 

 end of this paper. For some values of 6, ranging from 6 to 16, com- 

 parison is made between the successive approximations for e~^Ii{b) 

 and its true value as given by Watson in his Theory of Bessel Functions. 

 The last three columns of Table II give the figures for w = instead 

 of w = 1/2. 



Appendix I 

 When the expansion of <pu>(x){dx/dt) is not obvious we may proceed 

 as does Laplace in expanding idxjdt). 

 Equations (2) and (3) give 



(7) 



X = X + tg-\ 



This last equation, together with the Lagrange-Laplace expansion 

 theorem, gives for a function of x the expansion 



f{x) = AX) + L 



s=iS' L 



dx^ 



\ ) n^) 



. x=A' 



