22 Art. 4. — M. Knniyeda 



Hence we liave 



/(/) = r(r) cos {irTr)}."- (log ?.y-\i+e,) 



+ r(.9)cos(/-i-s-);r^'(i+0- 



Similar!}^ we can prove that 



Z(;.) = r(r)sin{h-7T)?r^ilog?.y-'(l + sn 

 + r(.s) sin (^-i5;r)A-^(l +£/")• 



Thus we obtain 



J"(/) = A"^.îc'-^ (log — )' \lx 



= Fi^r) eW^i/r' (log ;0"^ (1 + 



where 



lira £ = 0, lim £' = 0, 



r and s having the values of (22). 



12. This result may be verified as follows. 



Hard}^ proved* that, if B(r)>0 aiid B{^)>0^ then, for pure 

 imaginary values of t, we have 



= r(r)(-0-'[iog(-/)}'-Ui+^.) 



where 



(-^)-'- = exp[-Hog(-0] =exp[-r[log|^|-k.T7}], 



V = ex\y[ — s\ogf} = exp [ — s{logl^| + ^e;ri} J, 



*Proc. London Math. Soc. Ser. 2, Vol. 2, pp. 401 ai x('<i. 



