484 
ME. E. W BAEXES OX IXTEG-EAL FUXCTIOXS. 
or, changing cr into — oq, 
m — I 
S n a log 
n=l 
rrn cr CC ( — VR, fP l+l 
i « = j ^ lo g « dn - i'(- <r) - 2 log m + s ^TT « i<T log m - 
Thus 
Cy o 
717 log = £' ( — 
cr 1 
We have, therefore, 
log n ( 1 + ;y) = £(— a-) log 2 + pC{~ o-) + 2 ( ~~ l (- ps + cr) 
.,= 1 
<t> M 
]_ f 7 <7 + 1 <7+1 
, 1 f * < 
+ 7TlJ 2 
p t~p 1 clt 
1 + X' ( —)T 
s = -k 
The last integral is equal to 
<T+ 1 
[<f> ( m )] p 
P _J_ 4 f J 
" - 4>( m ) 
4 1 
cr + 1 
, + i T ii 
_ [ 
[ * J 
<j+1 
<b (m)]p 
cr A 1 
s + 
p _ 
_ 77 
cr +1 :r + l 
P IT Z P 
cr + 1 
. 7T . C7 -f 1 
Sill - 
_<f> ( m )_ 
cr + 1 cr + 1 
Sill 7T . 
Thus we have the asymptotic expansion 
II 
n=l 
7 , ^ 
lV + , f 
<T + 1 
7T - 
s , - z p 
?£(,-*) »?£(-*) + - f(-p« + <7) + sin :i +T +1 0-+ 1 
z e *=1 p 
We note that the first term of this product vanishes when cr is an even integer. 
§78. It is now possible to write down the expansion of 
where \x a is algebraic and of the form a 0 n' T + cq n a ' + a. z n a - + . . . , in which 
cr > cr 1 > (T.i > ... 
For such a function is merely the product of the 
CO 
« 0 th power of II 
n -1 
the cq th power of II 
H = 1 
and so on, 
