An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 162,

Gamma function expressed by an infinite product.

287

and M' a mean value of in the interval from 1 to oo; thus:
1 ~ - e_~ ~x~n-c
VIII) d I F -(a) 1 1 1 1
VI ) da2 a 2 (a+ )2 (a+1)2 (+)2 ( )2 (a+l)2
n=O
(a > 0).
This is a uniformly convergent series for every positive value of a;
integrating it between the limits 1 and a, the result is:
-n = 00 n = c/
X) cdIF(a) dI F(a) Y(1 a 4 =)(a-) IA )
X) da ( da )- (1+\n a+t,) (a ) 2J(I + it) (a + n)
for a=1 --- ( n - 0
(a > 0),
or:
dl F(a) he 1
n -- U
where C, denotes the value of (d — ) d -- r(-) therefore by VI) the
for a=-
value of the integral:
c-Y
~X) C - - e- Y i ) d - rF'(1).
0
Integrating equation IX) again between the limits a and 1, we
find:
1Z --, c
2T~il) =~n=Oc (1+12)(azr a
r( )=2j ++ ) +) (a -- n)
or:
A;,,(a - a~n)
'0 0
XI) rF(a) = r+ - 1 )+ (-1
The constant C can be eliminated: Putting a = — 2 this becomes:
Ir(2)= — 0 = (t + i - 0C);
multiplying this equation by a - 1 and subtracting from XI) we
obtain:
*) Writing: -- C = where however each
r 1 + n 1 -+ n
o o
of these series apart is divergent, - C is evidently the same number that
occurs in the Logarithmic integral (~ 137), as can also be directly inferred from
the connexion of this integral with formula X) above.