4(30 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
If s be negative, we have 
oo rm 
— A (f)' ( n ) = j <f)' (n) cIn — ^(f>* (n) + ...+( — ) 
71 = 111 J Vs 
2t + 2! dwi~ t+l ^ ^^ ‘ - 
where y s = °o , there being no term independent of m on the right-hand side. 
And 
7)1— 1 
2 log <f> (n) = log cf) (n ) dn - | log <f> (m) — f 
Bi +1 
'-!=1 
yo 
2t + 2 ! dm* t+l 
log <f> (m) -f■ .. 
Hence, in the limit when h = oo , 
rm 
log F (z) — (m — I) log z -- j log cf) (n ) dn 
Vo 
+ t 
8 = 1 
* T( _ )* 
y— 1 rm ( \s ~s rm 
hr I P(n)dn+ { -j^ I 
^ J Vi J V — 
— i j — log (wi) + 2 
* rv _ v-i 
( W» + 
S=1 
dn 
y-8 A 
(-y 
-S;r 
, - .n" \ 
+ 2 2 n r . i - log ^ <“> + 
s<F (m)_ 
v (->' 1 <F(»0 
(=0 
S=1 
I (-r 1 ^ 
Now, when the limiting values for Jc = oo of the summable divergent series 
are taken, 
2 
s— 1 
x-y 
-i 
(-) 8 
(b s ( m) + , 
SZ s r \ / g( ps (. n i)_ 
- log cf) (m) 
= log ( 1 + ) - log 11 + 
J 
(f> (m)j 
— log cf) (in) = — logs:. 
Hence, asymptotically, 
Yo 
k 
s=l 
-1 
f m cf) s (n) dn 
( m z'dn ~ 
J S 2 s 
L J v* ° * 
J y - S scf> s (n)_ 
= (»»- 4) log * - [»log <4 ( »)];: + j 
+ t 
deb (n) 
n — — 
yo <MA> 
.1 Jy» W*( w )J ?-*. 
s (-)' 
m ?i0 s 1 (n) 
lv y.- 
* 
# M + j 
J y—s 
zrn 
cf) s+1 (n) 
def) (n) 
= ~ i lo g z + [> log cf) (n)] yu + 2 [f (n) w] y , 
S=1 * •* 
pH™) ,7/ 7- rr<M>«) / s_1 Z^'ylrU) 
+ vM0f + 2(-H *«-** + ( 
j 6 (vA f s=l J 6 (y t ) * r 
' (yo) 
<4 (y>) 
I <f> (y — l) 
