Oscillating Dirichlet's Integrals. \0\ 



I 



Circle of convergence, one singular point only at z = I, being regular 

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



ichere g{x) = Viix)Y^ [hi^)]''- {h{3^)V'' , -4 = a e"', a > 0, q and a de- 

 note certain constants, and p and all o"' s arbitrary real constants, 

 then the behaviour of the coefficient an. as n->x, is determined 

 asymptotically as follows. 



(i) If q = \, a = 71, or the singidarity is of the type 

 then 



(ii) If 0<cq<\, a = (i-\.q)^^ q^. 1]^^ singularity is of the type 

 1 



f(^\ — ^ -a{s\ulqn-i cos lq-K)l{l - zf ( 1_ \ 



then 



liliere Â- = (1 +g) g'^a'^^. 



(iii) If < g <; 1, « = (3 — g-)-^, or the singularity is of the type 



f(^\ — 1 ^-a(sin:^g7t + icosè<Z7c)/(l-^f „ ( ^ \ 



then 



_p-i p-'^-ij 



a„^ — ^ (l + g)-<'-i+^>(ga)'^+5 n ^^''' gin) exp j^- [kn^^'-{hp-^)-]i~\^ , 



k being the same as in the case {ii). 



Published April 20, 1919. 



