Oecillatjno^ Dirichlet's Integrals. 29 



\r = —a, —1 <;a<;0, 



(22) 



(<; = « + mi, 0<«<1, m=^0. 



Then 



= I(/) + iZ(/), 



where 



1(A) = f\-Jlogiy~'e^''-s'og(i!^cosh^ ^^^ 



J \ X J X 



J \ X J X 



Now 



= !,(/) + 7,;/) 



say. JCviclently the integral I\J^) is convergent, if a<0. The 

 integral ^é.^) may be written in the form 



LI?.) =y"\l_:r)-«Ylog-J-y^^^^°s^«s [1/(1--)} cos }il-x)dx, 



and, when :^ is small, we have 



log^ = a-[l + 0(a-)}. 



Hence the integi'al I-i^^) is convergent if r/>0. 



The integral I\{'^^) may be divided into the two parts 



IXX) = (/"+/^'*)a;-'(log-I)"".-^i««-i^^(i/-'-)^^^rc 

 say, Ç being a sufficiently small positive number. Then 



./ X 



