Oscillating Dirichlet's Integrals. 97 



Now 



qA = qae^^-^''^'^\ 



and 





r~> e 



9/(1 + ï) 



Hence, writing p^ =p + q+l^ we obtain 



where /)i<cl + 2g. 



Thus, in the formula ((SO), the upper hmit of p is increased 

 by q. By repeating this process 7n times, the upper hmit of p in 

 (SO) may be increased by 7)iq. Thus we see that, in the formuhi 

 (80), p may take any positive vakie, whatever. 



Next, by (83), we have 



a{v + 1 , p) = -^ a{n, i? f 1 ) + ^ , aUi, p + q+\). 

 n + J n + i 



By repeated applications of this formula, the lower 1 inj it of p in 

 the formula (80), may be decreased as much as we please. Thus 

 we see that, in the formula (80), p may take any negative value, 

 whatever. 



Therefore the formula (80) holds for all real values of jd. 



The same argumicnt applies to the other two formulae. 

 Hence we can state 



00 



Theorem VIII. Let ^ a^z" be a power series, whose 



radius of convergence is unitij, representing a function f{z) which has 

 on the circle of convergence, one singular point only at z = 1, being 

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



f(z)=^^e'"^'-'y\A=ae''\ 



