Oscillating Diri chiefs Integrals. gg 



Proof. By liypothesis 



Hence, if we put ^ ~ ~2 '^^"'' 



we obtain 



¥ - " - 7 - 2 ' 



COS (« + g^) = COS [\7i + q{a' + (f)] = — siu g(«'+ ^), 

 and gs s g a'4- ^) s n- — ge, 



since — -ö-+e = f = Y~~^" Hence we have 



sin q[o.' + ^) > 

 as e > 0. 



Now, by (66), 



I /•/--r ^ I <r- 1 f^~\"n"' sin (/(«■ + ?) <^^ 



^q7irr'(y-ny^\lo ^ ^' 



and re-^-^-'^^'clt = ^ fV'"'-^' '''''' dt 



The last integral is convergent and tends to 



e-"du = 1 







as ri->0. Hence 



re-"''"'''''dt <:Kr{', 

 and we obtain 



