OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
449 
The expression 
e 
r(i+, o+, i—, o—) 
~ ,r ‘ (a+6) u a ~ l (1 — u) b ~ l du 
J c' 
has been denoted by Pochhammer by € (a, b) ; it has the advantage over the 
Eulerian integral ( u°'~ x (1 — u) h ~ l du of having a definite finite value for all values of 
Jo 
a and b. In € (a, b), the quantity 1 — u has the phase 0 initially at C, so that 
u — 1 = (1 — u) e~ m . The principal properties of € (a, b) are the following:— 
(1)' € (a, b) = £ (b, a), 
(2/ e (a + r, b) = (— 1 
a {a + 1) . . . {cc + r — 1) 
(« + 1) {a + b + 1) . . . (a + b + r — 1) 
€ (a, b), 
_ , 7 . , . (a + b — 1) ... (a -f b — r) , 
€ « - r, b) = (- 1 Y) - 1W * - 7 -7 € (a, b). 
x v ' (« — 1) (a — 2) . . . (a — r) v 
( 3 )' € (a, b) = — 4 sin an sin bn . E (a, b) 
when the real parts of a, b are positive, E (a, b) denoting the Eulerian integral 
f u a ~ l (1 — u) h 1 
J n 
du. 
which is equal to 
n (a -1)11 (b - 1) 
IT (a -f- b — 1) 
By means of (2) this theorem can be extended to the case in which the real parts 
of a, b are not necessarily positive, 
( 4 )' € {a, b) = € (1 — a — b, b) — € (a, 1 — a — b). 
We have 
r(l+, 0 + , 1 —, 0—) 
j u n+r (u — du 
f(l+, o + , 1-, 0-) 
__ c (*+*+])« u n+r ( 1 — u)~ n ~ m ~ l du 
hence, since 
— e (n+r)« e (n + r + 1, — n — m), 
€ (n + r + ], — n — m) = ( — 1 ) r ^ + ^ ^ + ^ ‘ ‘' ^ + - \ € (n + 1, — n — m), 
' x ' (1 — m) (2 — m). . . (r — to) v 7 
the expression ( 3 ) becomes 
MDCCCXCVI.—A. 
3 M 
