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.

~ 163. 164.

Convergent infinite product form of F(a).

291

Since this expression approximates arbitrarily to unity, merely by choice
of n, we have:
Lim Pm' Lim Pn.o*)
164. The convergence of the function gamma for all values
of a, for which no factor vanishes, is now demonstrated as follows.
Let us form:
1 _ aTT (1+-a_) a_1+a + ( a+-a) (l+ a)
2 _ v v v,. v e+
F(a) ma ('+ )'C(+
1 +
Writing the factor -- in the form 1 - n,, and applying
( u)a P
the Binomial series (~ 46), we have:
(+ ()a _+ a) _ a _+C a(a- ) (1 __ )_ (1 + )
16~ (1_ a )' (1
un 2~/ ~ n/1 \a ~~.2 \~/ l 1. n n/
or:.
a(a-1) (12 + 1)
(O-)- —,7 (O < o < < ).
1.2 (n +n 1(.
If a be positive - ( - is certainly a proper fraction,
and the series:
<.,, a(a6 - 1)/1 1,,
un + q1f++l +,+ +. * < - )(a j( + 1)2+ ( ) + )
converges absolutely.
If a be negative, we can choose n so large, that for any possible
value of 0, ~ --- shall be less than some determinate
1+~q (1 +number, ex.gr. less than 2, so that the series:. l ' n-' <,a(a - 1) 2,
Un + Jn+1+ n- + (2 - 1) ( + -L + 2 +)
1 2.2 n- n+- (nl + 2)2
likewise converges absolutely.
It is accordingly proved, that the infinite product,_, therefore
_ _ _ _ _ If (a)
*) Although infinite products were introduced almost simultaneously with
infinite series, the fundamental theorems regarding them were first proved by
Weierstrass: Ueber die Theorie der analytischen Facultiiten, Journ. f. Math.,
Vol. 51; reprinted in his: Abhandlungen aus der Functionenlehre', p. 183, 1886.
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