52 Mr Watson^ The limits of applicahility 



It follows that 



^^^^^A'^^'"'""'"'"'-^^^' 



as i — > /Lt + 0, where 



^^ -(r-l)!(r!)^ 



</><'•)(;.) 11 (/,<'•) (/.)!}--' 



Since j wF ] — > oo with h, by hypothesis, we deduce from Bromwich's 

 theorem that 



cos 



r p(-)o (If reoo 



^<*> sin «+ St '^ ~ "" '"' .1 . <^'>"'"' -n ^'^- 

 Writing %e s &>, we get 



re CO rcc 



I (%^)™~^ COS %<^% = e I &>'"~^ cos ctx^o) = eF (??i) cos -| 



and similarly 



(%e)'"~^ sin %c^% = F (m) sin ^7«7r. 



7?2-7r, 



In like manner, when t-^ /jl- 0, 

 dt 



and so, since 1 717 | — > oo with n, we have 



/, ^ (*) sm "^ 4 ^'^ ~ (-)•■ -'" ^■^i „ (^'''•""' sin ^<'^- 



Collecting our results, we see that the first approximation to 

 / is 



/ ~ [{AKe + (— )'■ AiKr]} cos ^iutt cos {nfM-f (fM)] 

 - [AK + (-)'• A^K] sin |m7r sin {^i/^y (/i)}] 

 = A [cos (w^-/' (/u.) + ^emir] + J.i cos {ufx^f (fi) + ■|?;??i7r}] 



(r - 1) ! (r !)'»-! F (m) 

 ^ 27rw'^{|<^<'-'(;u)j}'" ' 



and this is the result stated. 



The formula fails to be effective in the neighbourhood of those 

 values of n for which the expression in [ ] vanishes, as the error 

 in the approximation then becomes comparable with the approxi- 

 mation obtained. 



[It is evident that if the cosine in the integral defining / may 

 be replaced by a sine, then the cosines in the approximation are 

 replaced by sines.] 



