360 BELL SYSTEM TECHNICAL JOURNAL 



or 



I,,{u)|^[^{Y4NY = ± (l/2ViV)2'"(^2^/m!), 



<• OT=0 



where, writing D for the differential operator d/dx, 



Aim ^ [/)^2mg-(2m+i) cos {x2u^fN)^^^x 

 or, by the Leibnitz theorem for the product of two functions, 



A2m = L (^^)(2u4Ny COS (u^2N + r7r/2)[i).2.,-.^-(2.+i)-]^^^ 



and, therefore, 



^ = cos (W2iV) £ (^27) ' " '''';^^^^" [J,'-^-,-"-^"]„.v 



- sin (urn) £(2,^!ri) ''"^^"if^"' [^.--'r--].-.. 



on separating the even and odd terms in r. Now setting w — r = 5 

 and summing with reference to 5 and r, instead of m and r, gives 



= COS („V2iV) L (2 v]v) Ml So "WT- (7+7)-! 



X [5...,-<.«.«,]„, - sin (WSV) E (^)""" (27^ 



" (2r + 25)! (— 1)'"Z<^''+^ p 2«-lp-(2r+2s+l)-| ^ 



But, writing w/g = z^, we have 



i-oV (20! ){r + s)\^-^^^ J 



= (- l)»M-(2s+l) 



h {r + s)\ (2r)! 



00 C_ iy+»j,2r+2j 



,fo (r + 5)! 



= (- l)-/^-(2»+l)[;i)^2,^2.+l_p^2.g-v2-j^ 



since D-^'v^"^ = for m less than 5. 



