Oscillating Dirichlet's Integrals. 3 



As for the cosine-integral C(>^), it will be seen that, in the two 

 cases (A) and (B), it alwa3^s tends to zero as ^i-^oo, when conver- 

 gent, and, in Case (C), its behaviour is very similar to that of 

 the sine-integral. I have found for it asymptotic formulae 

 whose range of validity is the same as that of the formulae for S(X). 

 That the formulae should cease to hold when the order 

 of the integrals sinks as low as -j- is to be expected. For, it 

 is easy to see that the parts of the integi'als away from x—0 are in 

 general of the order -r-, so that in such cases the behaviour of S{^) 

 or C{À) is no longer dominated by the parts near x=0. 



3- The principal results arrived at are as follows: Writing 



oix) = x-'^e^x), x'<e< (i/x)', 



in Case (A), 



S(X) ^ -r(-a) sin (^an) p(lß) e"^'^'^ ( - 1 < a < 0), 



S{X) ^ ^7rp{\ß)e^Wiy (a=0, p<l). 



Combining these results with Theorem A, we obtain the 

 theorem: 



Ifl<a<l(llx), p<<t' and 



/> = x-«6> {x), x' <e< {l/xY, a^ 1, 



then we have, as ?.^-co , 



(3) 



(Sf(;0 = O(/-i+*) (a^-l). 



S{X) r^ -r{-a) sill (Utv) p(\/k)e"^''^ (-l<a<l), 



s{ä) ^ / r(i/;.) (a=i). 



where T{x) = f p{t)e''^'Ht. 



/ 



III the particular case a=0, the factor —r'{—a) siu {^ait) is to be 

 replaced by its limiting value 2^. 



The corresponding formalae for C(A) are as follows: — 



