Oscillating Diri chiefs Integrals. 



13 



8. Now we pass to the cosine-integral 



G (;0 = /' ' o (x) e ''^'^ ^^^^ dx. 



Ja X 



The integral 



./ X 



is convergent if, and only if, 



(18) p<xa' 



as 07-^0. This involves P<^, as <?'<—. As the factor cos /la; of 



the subject of integration of C{)) ig ultimately monotonie, this 

 condition (18) is the necessary and sufficient condition for the 

 convergence of the cosine-integral C{^^). 



Performing integration by parts, we obtain 



C{X)=.riLe-J-(^^\dx 

 J X ax \ A J 



(19) 





where 



(20) 



Jo X 



J 



;- sin Xx 



B, =^-p\ B,= oo'. 



The integrals J^'^ and /^'^ are of the type of S{).). 

 As before, we can write 



f) = x~" 6 (x), 



