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.

288 Examples on the calculation of definite integrals. Bk. III. ch. VII.?z = CrD
rF(a) = ((a --- 1) 1 +n +-
1 '1n 1+ nz
n=0
m-=l
= (a - 1)1 (I+ m- (1 +t - ) '
or:
(,,t /+ l ~a-l-I
XII) F (a) = ' 1 +)
Accordingly, when we pass over from the logarithm to the number,
we have [(a) expressed for calculation by an infinite product:
Im (m+l t)a- 1.2.3... m11(m l)a - 1
F(a) (ac -+ -1) - - (a+ i) (a + 21 )... (a+ m-1)
m-1
(0 + l)a-1l
a t a)(I+ -a) ) a( +z - )
(for mn --- o),
or, as this expression can be written, when mn is replaced in the
numerator by the value (m + 1) (l - - ), and it is remembered
that as mn increases arbitrarily, the factor 1 --- and likewise
m +1
(1+ -+ differ inappreciably from unity:
XIII) r (a) = ( - + a)a i a.
a). (, 2(+=a -.
m==2
(mn oo).
Gauss*) employed this formula as the definition of a function
and derived all its properties from this infinite product. This can be
shown to converge for every finite value of a, which does not make
a factor of the denominator vanish, so that this definition is more
comprehensive, than the Eulerian integral.
163. We are going to prove this, by answering in general the
question: Under what condition does an infinite product converge?
This is an important question; for as was indicated in ~ 38 and is
here worked out for a definite function, the formation of an infinite
product is a second instrument for the expression of a function, not
symbolically, but suitably for numerical computation. The following
*) Gauss: Disquisitiones generales circa seriem infinitam. Werke Vol. III.