490 
MR, E. W. BARNES ON INTEGRAL FUNCTIONS. 
Thus 
/M _ Io § g _ 
1 £ pqm q 
1 1 * 
m + + - t' 
PI I =-* 
'( — )’- 1 / g? M 
1 4 ' r (-) 8-1 
I s=-Jc _ P + s \ 2 . J 
log 2 | 1 | 1 J 7T , 
— m — + - i / 
2 
PI 
Z \P 
Sill 7 Tp 'P 
— 1 — log — 
Therefore 
Again, M is given by 
/(*) = 
77 zr 
pq sin 7 rp 
a am 
e — G 
1 - e“ 
p + Lt + 
r=co « 
l-o + 
/,• ( vn ,,2;+2' 
^ (, j Of+i <* 
<=0 
2 ^ + 2 ! 
• • (1), 
dm 
e a 
= M -(- — . --- , when a = pq. 
a 1 _ e 11 
Tlius 
M = 
eji 
1 - en 
Also — Fj is given by putting a = p + 5 q in this same expansion. 
Thus 
F — 
e? . r+- ! 
1 — e q _ p + s 
Again, by substituting a + e for a, expanding in power of e, and equating 
coefficients of the first power of e in the asymptotic identity (1), we readily find 
evi 
— log F„ = q —-— . 
& 0 1 (1 — e^y 
If, then, p is not an integer, we have the asymptotic expansion 
n 
n = 1 
i +A 
1 a n j 
. r pqn vnn . „ . 
e ' 1 c l 1 p ( —z \m 
m= 1 
(w 
V'l 
= z Qpq sin irp (l-e**)*’'' M jL p T7).< 
§ 84. Suppose next that p is an integer—so that p = p. The analysis will, of 
course, he slightly more complicated. 
The constant F_„ will be given by 
m— 1 
Lt \ S j _ + d n _ F_ p _ ] 
s=p L »=i j J 
or 
m — l = m — \ — F_p, so that F_ ; , = i 
