EXPANSION FOR FUNCTIONS OF HIGH ORDER 

 Likewise 



" (2r + 25)!(- 1)'-m2'-+i 



361 



Therefore, finally, 



r/)^2«-lg-(2r+2s+l)l 



o4tb = ~^ ("^2^' Uiy)" c«-"-"'^'j(- ')■ 



where 



(25 - l)!ii:2.-i = \:D,'^-'v'^e-^'H2s-i(v)2.=x. 



Substituting sin (x2u-y[N) for cos {xlw^N) in the equations defining 

 Aim and then proceeding exactly as above we derive the corresponding 

 expansion for In'{u). 



V 



To obtain the values of K^a and i^2s-i note that 



Xg^=\- {x- X)/3X + (x - X)V4X2 - {x- Xf/SX-' H 



gives, for a; = X = 1/V2, 



g= V2, 



(Zg/c?x == — 1/3, 



d^g/dx^ = (4V2)/9, 



d'g/dx^ = - 88/45, 



^V^x^ = (824V2)/135, 



d'g/dx^ = - 28184/567, 



V = (1/V2)m, 

 (/t'/c?:)i(; = (1/6)m, 



c?V^x' = - (1/3\^)m, 

 <^Vrfjc» = (53/90)m, 



<iV<fx4 = - (211V2/135)m, 

 d'v/dx'' = (79/7)u, 



etc. 

 Therefore 



36V2(w-3J^2) = e-^^'^Cw^ - 6w^ - Qm^ ^ 12), 

 7776V2(M-5i^4) - e-^"^(M^2 _ i2i/io - 183.6«« + 1432.8«« 



+ 2889m^ - 10368^2 + 432), etc. 



6(u-'Ki) = we-5"'(M2 _ 3)^ 



