19. The gamma function.

The idea of "interpolating" a sequence $(u_n)$ by the values of an integral dependant on a real parameter $\lambda$ and equal to $u_n$ for $\lambda = n$, goes back to Wallis (cf. pp. 187 ff.). It is this idea that was the main guide for Euler when, in 1730 ([108 a], (1), v. XIV, pp. 1-24), he has the idea of interpolating the sequence of factorials. He starts by remarking that $n!$ is equal to the infinite product $\prod_{k=1}^{\infty} \left( \frac{k+1}{k} \right)^n \frac{k}{k+n}$, that this product is defined for every value of $n$ (integral or not), and that in particular, for $n = \frac{1}{2}$ it takes the value $\frac{1}{2} \sqrt{\pi}$ from the formula of Wallis. The analogy of this result with those of Wallis leads him then to take up again the integral $\int_0^1 x^e (1-x)^n dx$ ($n$ an integer, $e$ arbitrary), already considered by this latter. Euler obtains its value $\frac{n!}{(e+1)(e+2)\cdots(e+n)}$ by the binomial expansion; a change of variable then shows him that $n!$ is the limit, for $z$ tending to 0, of the integral $\int_0^1 \left( \frac{1-x^z}{z} \right)^n dx$, whence the "second Eulerian integral" $n! = \int_0^1 (\log \frac{1}{x})^n dx$; by the same method and the use of the formula of Wallis, he obtains the formula $\int_0^1 \sqrt{\log \frac{1}{x}} dx = \frac{1}{2} \sqrt{\pi}$. In his later work, Euler returns frequently to these integrals; he discovers in this way the relation of the complements ([108 a], (1), v. XV, p. 82 and v. XVII, p. 342), the formula $B(p, q) = \Gamma(p) \Gamma(q) / \Gamma(p + q)$ ([108 a], (1), v. XVII, p. 355), and the particular case of the Legendre-Gauss formula corresponding to $x = 1$ ([108 a], (1), v. XIX, p. 483); all of it of course without worrying about questions of convergence.

Gauss continued the study of the $\Gamma$ function in connection with his research on the hypergeometric function, of which the function $\Gamma$ is a limit case ([124 a], v. III, pp. 125-162); it is during this research that he obtains the general formula for multiplication (already noted by Legendre not long before for $p = 2$). The subsequent work on $\Gamma$ has mainly been concerned with the extension of this function to the complex domain. It is only recently that it has been noticed that the logarithmic convexity property characterised $\Gamma(x)$ (in the real domain) up to a factor amongst all the solutions of the functional equation $f(x+1) = x f(x)$ ([26], pp. 149-164); and Artin has shown [7 d] how all the classical results about $\Gamma(x)$ can be simply linked to this property.