The Gamma Function

Abstract

The problem of extending the function n! to real arguments and finding the simplest possible “factorial function” with value n! at n ∈ ℕ led Euler in 1729 to the Г-function. He gave the infinite product

$$\Gamma \left( {z + 1} \right): = \frac{{1 \cdot {2^z}}}{{1 + z}} \cdot \frac{{{2^{1 - z}}{3^z}}}{{2 + z}} \cdot \frac{{{3^{1 - z}}{4^z}}}{{3 + z}} \cdot\ldots= \prod\limits_{\nu= 1}^\infty{{{\left( {1 + \frac{1}{\nu }} \right)}^z}{{\left( {1 + \frac{z}{\nu }} \right)}^{ - 1}}} $$

as a solution.^l Euler considered only real arguments; Gauss, in 1811, admitted complex numbers as well. On 21 November 1811, he wrote to Bessel (1784–1846), who was also concerned with the problem of general factorials, “Will man sich aber nicht... zahllosen Paralogismen und Paradoxen und Widersprüchen blossstellen, so muss 1∙2∙3.... x nicht als Definition von Пx gebraucht werden, da eine solche nur, wenn x eine ganze Zahl ist, einen bestimmten Sinn hat, sondern man muss von einer höheren allgemein, selbst auf imaginäre Werthe von x anwendbaren, Definition ausgehen, wovon... jene als specieller Fall erscheint. Ich habe folgenden gewählt

$$\prod x = \frac{{1.2.3 \ldots k.k^x }} {{x + 1.x + 2.x + 3 \ldots x + k'}}$$

wenn k unendlich wird.” (But if one doesn’t want... countless fallacies and paradoxes and contradictions to be exposed, 1∙2∙3... x must not be used as the definition of Пx, since such a definition has a precise meaning only when x is an integer; rather, one must start with a definition of greater generality, applicable even to imaginary values of x, of which that one occurs as a special case. I have chosen the following... when k becomes infinite.) (Cf. [G₁], pp. 362–363.) We will understand in §2.1 why, in fact, Gauss had no other choice.

Also das Product 1∙2∙3... x ist die Function, die meiner Meinung nach in der Analyse eingeführt werden muss. (Thus the product 1∙2∙3... x is the function that, in my opinion, must be introduced into analysis.)

— C. F. Gauss to F. W. Bessel, 21 November 1811