of the Principle of Stationary Phase 51 



wlien n is large, is 

 r- 1) !(r !)'"-! r(m)[^ cos {nfx,^ f'(fi)+ ^em-n-] + A, cos {nfj:'f'{fM) + 1 7?m7r}] 



provided that < 1 — X < r ; wAere e = ± 1 according as bt — tf{t) 



increasinq , . , , ^ ,. , 



%s an , . f unction when t > a, ana ?? = + 1 accoratnq as the 



decreasing n • - 



same function is . . when t < a. When n—^cc hti only 



increasing 



such values that cos [nfM-f {/u.)} is always zero, A, may lie in the 



extended range —r<l — X< r. And, finally, F{t) and bt — tf(t) 



inay be infinite at t= a, ^, provided only that the integral converges 



for all sufficiently large values of n. 



3. For brevity, write tf (t) ^ (f) (t). Then ^ is given by the 

 equation 



b-<f>'(,M) = 0, 



so that, when t — fi, is sufficiently small, 



bt - tf(t) - t(}>' (/jl) -(f){t) 



= {/.f (/.) - </) (/.)} -(t- fxyr' {t')/r !, 



where, by Taylor's theorem, t' lies between fx, and t. 

 Now define a new variable yjr by the equation 



bt-tf{t) = fi<l>'{ix)-c},ifi,) + f, 



and let 7, F be the values of yfr corresponding to t = a, t— /3. 

 Noticing that /x^' (/j,) — (f> (fi) = fi^f (/n). we have 



J. cos {n/M'f (fi)} r^„,,, , u 



sin{n/jb^f' (a)} T^et/^x • , 7^ 

 Zir J a 



i/r being a monotonic function of t when a ^t ^ /x and also when 

 fi^t^^. 



Now e, 7; have been so chosen that e-yjr and rj^jr are positive 

 when t > /M and ^ < //- respectively ; hence, when ^ —>//, + 0, we have 



df 



ef '^ {t- fiY </)<'■' {fi) 1 ^ r !. 



