Oscillatiag Dirichlet's Integrals. 99 



where A = a e"', a>0, ir ^/ < 27r, q > 0, 



and p, r denote arhitraiy re;il constants. 



First of all, in considering the integral I[ri), we observe tlvat 



log T = log ( - — «"'"M = loo- — + <fi. 



Hence, without ditlicuUy, we can ])rove that Lemma 11 holds 

 also in this case; namely, if 0<;(/si, (i + g)_^s « s(3-g)-^ and 

 p<l-+q, then 



lim J(ri) = 



)-l->0 



Next, when 6 is small, we have 



log ,. = log ■ ■ , ,y+(i"-|^)^ 



1 —e^ 2 sin U' 



= logi- + i-/+0(^) 



= log|{l+o(llog|)), 



'<>B-x^.)"=(<'4n'+«("°4)}- 



whence 



Similarly 



(,og^_U.y = (,o,]7(i+o(i/io4)}. 



Hence the dicussion may he carried out quite similnrly as in the 

 preceding case, the presence of the logarithmic factor producing 

 no great change in the an;dysis, and \ve content ourselves with 

 giving only the following results. 



CO 



Theorem IX. Let Z^a,^z" he a power serie.^, whose radms 



71 = 



of convergence is vriity, 7'epresenttn(/ a function f{z) n-hich has, on 

 the circle of convergence, one singvlnr point only at z — \, being 

 regular at every other 'point on it. If the singularity is of the type 



