Oscillating Dirichlet's Tnteo-rals. Ç3 



II Case in. which the Snigiihtriti/ /.< of 



t^<^ Type n^^'-""'^' 



47- Let us consider the case in wliicli the function f{z) has a 

 singularity of the type 



Avliere A = ae'"\ 



and p, g, a, a are real constants such that 



■p >, =, <0, a > 0, q > 0, s r/ < 2-. 



It is to be understood that, when p and q are not integers, 

 (l — zY and (l—z)'' assume respectively the values 



where log(l— ^^) assumes its principal value. 

 At first we consider the integral 



^w=iys?i.7^*- 



Let F be any point z = re"' on the arc DPB and let f denote the 

 angle between the radius OA and the straight line AP, namely 



(f = ^ OAF. 



Then \ — z = 1 — r cos d—ir sin 6 = y^ cos ç- — i)\ siu <f 



= — -e' 



(i-zy 



7T^ := JLe(«+..H- = ^ (cos ,a + qç) + i sin (a + qç)], 

 y(^) = ^— e« COS (a+g?)/?-/'' _ g [p^ + a sin (« + </=);'•/'] ; ^ 



dz = ivie ^* (ff . 



