EXPANSION FOR FUNCTIONS OF HIGH ORDER 



359 



tories, for the computations involved in the preparations of Tables I 



and II. 



TABLE II 



IV 



A simple change of variable gives 



(5) /„(m) = {4N)^+^ {e-'^'^xY cos {x2u^N)dx, N = 2n, 



Jo 



(6) I„'(u) = {4NY+^ {e--''xY' sm {x2u4N)dx, N = 2n + \. 



Jo 



Set y(x) = e-^lr, and note that dy/dx = for jc = X = l|^[2'. Now set 

 Y = y{X), 



lg{x)J= {\ogY-\ogy)Kx-Xy 

 1 



X' 



{'^Hi'^r-im'^-i 



(7), t= (x- X)g{x). 

 These transformations give 



e-''''\dxldt) cos (x2Hy[N)dL 



■00 



By (7) and the Lagrange-Laplace expansion for a function of x in 

 powers of t we obtain 



Xoo 

 6-A^'-[/2M2,„/(2/;/)!>// 

 00 



