Historical note

(N.B. Roman numerals refer to the bibliography to be found at the end of this note.)

The idea of “interpolating” a sequence $(u_n)$ by the values of an integral depending on a real parameter $\lambda$ and equal to $u_n$ for $\lambda = n$ , goes back to Wallis (III, p. 55). It was this idea that principally guided Euler when, in 1730 ((I), v. XIV, p. 1-24) he set himself to interpolate the sequence of factorials. He began 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},

which product is defined for every value of

n

(an integer or not), and that in particular, for the value

n = \frac{1}{2}

it takes the value

\frac{1}{2} \sqrt{\pi}

by Wallis’ formula. The analogy between this result and Wallis’ led him then to revisit the integral

\int_0^1 x^e (1-x)^n dx

(n

integral,

e

arbitrary) already considered by the latter. Euler obtained the value

\frac{n!}{(e+1)(e+2)\ldots(e+n+1)}

for it, using the Binomial expansion; a change of variables then showed him that

n!

is the limit, as

z

tends to 0, of the integral

\int_0^1 \left( \frac{1-x^z}{z} \right)^n dx,

whence the “second Euler integral”

n! = \int_0^1 \left( \log \frac{1}{x} \right)^n dx;

by the same method, and using Wallis’ formula, he obtained the formula

\int_0^1 \sqrt{\log 1/x} dx = \frac{1}{2} \sqrt{\pi}.

In his later works Euler often returned to these integrals; he thus discovered the complements’ relation ((I), t. XV, p 82 and t. XVII, p. 342), the formula

\mathbf{B}(p, q) = \Gamma(p)\Gamma(q)/\Gamma(p+q)

((I), t. XVII, p. 355), and the particular case of the Legendre-Gauss formula corresponding to

x = 1

((I), t. XIX, p. 483); all this of course without worrying about questions of convergence.

Gauss pursued the study of the $\Gamma$ function in connection with his research on the hypergeometric function, of which the $\Gamma$ function is a limit case (II); it was in the course of this research that he obtained the general multiplication formula (already noted by Legendre a little earlier for $p = 2$ ). The later work on $\Gamma$ was mainly concerned with extending this function to the complex domain. Only recently has it been appreciated that the property of logarithmic convexity characterises $\Gamma(x)$ (in the real domain), up to a constant factor, among the solutions of the functional equation $f(x + 1) = x\, f(x)$ (III); and Artin showed (IV) how one can link all the classical results on $\Gamma(x)$ simply to this property. We have followed his exposition quite closely.

(I) L. Euler, Opera omnia, Leipzig-Berlin (Teubner): t. XIV (1924), t. XV (1927), t. XVII (1915) and t. XIX (1932). (II) C.F. Gauss, Werke, t. III, Göttingen, 1866. (III) H. Bohr und J. Mollerup, Laerebog i matematisk Analyse, t. III, Kopenhagen, 1922, p. 149-164. (IV) E. Artin, Einführung in die Theorie der Gammafunktion, Leipzig (Teubner), 1931.