44 Ml' Watson, Bessel functions 



then it is fairly evident* that when ^0 ^ir (i.e. when ^ ^ tt), 

 fi{<f>) is bounded and has only a finite number of maxima and 

 minima (and therefore it has limited total fluctuation). Con- 



sequently, since f -^jr ■• sim/rc/i/r is convergent, we have;]: 

 . 

 v 77 r ^ 



Lim n~i ^ (f)"'^ sin (n(f>) . f\ {(f)) d(fi =/i (0) | yjr'" sin yjr d-yjr. 



Therefore, since /i(0)= 1, we have 



n-ijy~fsm(ncf)).f,{cf>)d<f> = ^T(V> + o(l), 



and so Jn(n) = 2~ ^ S~ '"^ ir-^ T (^) n~ ■- + o (n~ ^). 



To obtain the second approximation to Jnin), we obser^ 

 that, when 6 is small, 



(6«^)^sin^ (i-i&^+^e^-...){i-^ ^e-^+ ^L^e^- ...f 



Consequently, if </) " ^' { (1-^0^6'^ ~ efj ~ "■^' ^'^^' 



we have /o (0) = 6 ~ ^ ^ 35, Also, as in the case of/i (</>), we assume§ 

 for the moment that /2(^) has limited total fluctuation in the 

 range (0, tt). The application of Bromwich's theorem is therefore 

 permissible, and we deduce that 



Lim ni ["(^ - -^ sin (?i</)) ./, {j>)d<l> = 3^ 2 " ^- T (|)/35, 



M-s-oo J 



* A formal proof will be given in §5a that /j (^) is, in fact, monotonic and 

 decreasing (we use the term decreasing to mean non-increasing). 



t Euler's result that / ip''^^~'^ sin \p cl^p — V [m] sin {\mir) , when -\<m<l, is 



well known. 



J Bromwich,I?i/ini<e Series, p. 444, proves that, if f{<p) has limited total Jiuetua- 



f ^ sin Hd) 

 tion in the range (0, h), where 6>0, and if U,^— I — - — f((p)d(p, then 



H-^ao «-*oo J W J f 



but his analysis is equally applicable to the more general integral 

 V^=n'"' i (p'^-'^ sin {7i4>) . f (^) d(p (-l<m<l), 



and hence 



Lim F„=Lim I i/^^-i sin i// ./(i///?i)(7i/'=/(0) I f-^ sin xj^ df. 

 rt^-x M-*-Qo Jo J i) 



% A formal proof will be given in § 5 b that/^ (<p) is monotonic and increasing. 



