Oseillatiuy,- Dirii-lilet's Intt><;-rals. 87 



(()4) I{v) = J(n)+J[n). 



49- Integral J(n). First of all, we shall consider the 

 integral J(n). 



Observe that, when d is very small, we have 



COS [a + Uj[7: — âj] = cos (a + hj') [1 + 0(6'^)} 



+ sin (a + hl7T).hlO{l + 0(6')], 



sm [a + hqi- - 6)] = sin (a + ^q7r)[l + 0(6')} 



- cos (a + l-q7t). h2d[l + 0(^)1 . 



Hence the equations (70) may be written in the forms 



r(-ß\ = i a + 0(d-) ^ e*^"' ^'^"^ <«+*'/") + ^'^"^ «i^ («+ è'7^)] + o(e-"0^ 

 (70') 



</'{^) = M^-^) + js[sin(« + ig7r)-ig^ cos(a4-è-g^); +O(r"'0- 



Under the conditions (OS), the discussion may I)e divided 

 into the four cases 



( i ) q = 1, u = 7z, p < '2, 



(ii) 0<q<l, a = (l + q)—, p < 1 + q, 



(iii) < q < I, a = (3-qy^, p < l + cy, 



(iv) < q <1, (l + q)^<a<(3~q)^, p < l + q. 



50- (i) The case in which q = 1, «;= tt, _p < 2. 

 In this case 



« + jq- = ^rr. 

 Hence, by (70), 



