g Art. 4. -M- Kimiyecla : 



While I was working out this paptr at Cambridge, Hardy, being 

 struck with this fact, tried to attack the problem witli a quite 

 different method and obtained tlie following result*: 



then 



(12) a„ ^ -77orr^\ T^^«)" '^^^^ ' 'M "^(^^o'^n^V 



It is very probable that formula (12) also holds fur complex 

 values of a. If, in this formula, we replace a by 



A = ae"', 

 where 



« = (l + g)-^ or (3-g)-|-, 



then we get (10) and (11). This shows that (12) holds also when 

 a takes these special complex values. 



PART I.t 



Oscillating Dirichlet's Integrals 



I. Division of the Problem into three Cases. 

 5. We have to consider the integrals 



(1) S{X) =1 \,{:x) e^'^-^^^dx, 



•''0 X 



(2) C{X) = r'p{x)e''^-^^^^^dx, 



'' » X 



where p[x) and '^(a-) are L-f unctions and <t>1 as a-->0. As was already 

 mentioned, these integrals will be called " tlie sine-integraP' and 

 " the cosine-integral " respectively. It will be supposed that ^ is 

 a positive number so small that the range of integration does not 



* Me.isent/er of Math. Vol. XL VI (1916) pp. 70 — 73. 



t A preliminary notice of this Part appeared in the (,>uarti'rli/ Journal, Vol. XLVIII (1918) 



pp. 113-133. 



