48-2 
MR. E. W. BARXES OX INTEGRAL FtTXCTIOXS. 
We shall find it convenient to put 
rr/o 
/ u,„rhi = x( m )’ so that I jj.„dn — x(ffo) = ~ M (say). 
And now, if the limiting values when k = =o of the summahle divergent series he 
' o o 
taken, 
log F (z) — log z 
| jjL„ dn d - dl 
"Til 
log <f> (n) dn + 
^ Vo 
m 
(n) dn — \ ( m ) 
7' 
^■S 
c-~ 
r t 
=£ 
f^?n 
_! y _. 5 <p s '0d 
2 A( m )_ 
+ 
(-y 
r»/+l 
f7 2/+1 
f=0 
'it + 2 ! dm~ t+l 
/x,„ log (p (m) — fi m log z + 
/. 
v 
S=1 
(—y 
jwhl 
A C m ) 1 
The last term vanishes as for the corresponding case of non-repeated functions. 
After reduction, we have 
log F 
= loc 
Cm “| cm 
I jjL„ dn A M —| ji, log (f) (n) dn 
' 7o 
i- ( _v-i r r m ,, i 
+ S { f'" M (/,) dn - 
*=i ■<* [J yt A J y _ s & (» 
(d 
d/z 
= M log z A fjt,, log r/> ( 77 ) dn + 2 
t (-) s r*‘ 
=1 sr 
(n) dn 
r Mm) x [A(0 ]dt 
The last integral 
A, (— yt* 
1 A 2 - 
= U f x [* (- **)] i # ? n -Ay' * = /<*) say. 
If then we put log F 0 -= n„ log <f> (n ) c/ 77 , F. v = — | (n) dn, so that F 0 and 
Fj may he called the zero and s th Maclaurin constants for /x„ and (f>(n), we shall have 
the asymptotic approximation 
n 1(1 A - 
*=1 L\ Oh 
\P« 
/(-")+ i 
(-) s -h 
= FnS M e S=1 
