796 
MR. J. W. L. GLAISHER OR RICCATI’S 
28. The method by which the formula (7) was obtained in art. 20 is not satisfactory 
for two reasons, (i) because the integral \q x n ~ l e~ x clx is infinite in value when n is 
negative, while the gamma-function, winch is supposed to satisfy the equation r(n-bl) 
=nV(n) for all values of n, is finite when n is negative, except when n is a negative 
integer, so that we are not entitled to assume that we may always replace the integral 
by the gamma-function, and (ii), because it is assumed that we may change the sign 
of n in the equation giving the value of the integral. The following demonstration of 
the formula (7) is, however, I believe, quite rigorous. 
The gamma-function is supposed to be defined by the equation r(??) = x' l ~ l e~ T dx 
from n— 0 to n— 1, and by the equation l) = nY(n) for all other values of n. 
This is in effect the definition of the gamma-function generally adopted in analysis. 
We have seen in art. 20 that 
y— j" x~ x e~ x ~ x "dx 
7 J> jr 7 
satisfies the differential equation -— —4y=0, so that y=AM-f-Ba"N; and by 
a simple transformation of the integral, it follows that 
where 
H=1-~q(2a/3) + —— 1 
[ x l 1 e ai " x ~dx= Aa . 
Jo 
(2 «/3) 3 1 (2 a/3) 3 
(17), 
K„=l 
n -2 
1 
~n + 2 
(n—2){n — 4) 2! (n—2)(n~4)(n— 6) 3! 
■&c. 
(2<xfi) + 
(2 a/3) 3 
+ 
fii + 2)(u + 4) 2! (?i + 2)(?i + 4)(?i+6) 3! 
(2a/3) 3 „ 
Suppose n to be intermediate to 0 and 1. Put (3=0 in (17) and we have J 0 *V l e ^dx 
= Aa~ in , whence A = |-r(^w). 
Now by actual differentiation of the series represented by IT,, and K„ we find that 
d / , tt \ . _i ,_ t t-t d~K. n 2/3 -j-r 
(a n )=—\ncL * l 1 H„ +a , — =--Jv„ A0 . 
doL 
T 2 5 
and similarly 
<m n 
d/3 n 
9,, 
-H 
_9 2’ 
d/3 
da n + 2 
((3HQ = ±n(?'^K n _ 2 . 
Transforming the integral in (17) by assuming x= ,, it becomes 
f x~"~ { c~ p **~ pi dx= Aa -i "H„ + B/3 K,.; 
J o 
