Proposition 2

For every $x > 0$ one has

(Gauss' formula), and (9) where $\gamma$ denotes Euler’s constant, and the infinite product on the right hand side of (9) is absolutely and uniformly convergent on every compact interval of $\mathbf{R}$ not containing any integer $< 0$ (Weierstrass’ formula).

The function $\Gamma(x)$ is indefinitely differentiable on $]0, +\infty[$ and one has

and

the series on the right-hand sides of (10) and (11) being absolutely and uniformly convergent on every compact interval not containing any integer $\leq 0$.

Indeed, the proof of prop. 1 of VII, p. 306, shows that

whence Gauss’ formula, since $\frac{n}{x+n}$ tends to 1 as $n$ tends to $+\infty$. One can also write

so (in the notation of prop. 1)

and the series with general term $\log \frac{n}{n-1} - \frac{1}{n-1}$ is absolutely convergent and has sum $-\gamma$, where $\gamma$ denotes Euler’s constant (V, p. 242), whence we obtain Weierstrass’ formula.

For $|x| \leq a$ one has $|1/(x+n)^k| \leq 1/(n-a)^k$ once $n > a$, so the series on the right-hand side in formula (11) is absolutely and uniformly convergent on every compact interval of $\mathbf{R}$ not containing any integer $\leq 0$, for any integer $k \geq 2$; the same argument applies to the right-hand side of (10), since $\left| \frac{1}{n} - \frac{1}{x+n} \right| \leq \frac{a}{n(n-a)}$ for $|x| \leq a$ and $n > a$. Since these series are obtained by differentiating the series

term-by-term, and this converges for every $x > 0$, the series with general term $\frac{x}{n} - \log \left( 1 + \frac{x}{n} \right)$ is absolutely and uniformly convergent on every compact interval contained in [0, +∞[, and one has the relations (10) and (11) of VII, p. 308, for every x > 0 (II, p. 52, th. 1). Moreover, for every x ∈ $\mathbf{R}$, the expression $\frac{x}{n} - \log \left( 1 + \frac{x}{n} \right)$ is defined once n is large enough, so th. 1 of II, p. 52, again shows that the infinite product in the right-hand side of (9) (VII, p. 307) is absolutely and uniformly convergent on every compact interval containing no integer $\leq 0$.

The function $\Gamma(x)$, defined for $x > 0$, can be extended to the whole set of points x different from the integers $\leq 0$ so as to satisfy equation (7) of VII, p. 307, on this set: it suffices, for $-(n+1) < x < -n$, to put

By prop. 2 of VII, p. 307, the formulae (8), (9), (10) and (11) of VII, p. 307 and 308, with $D^k(\log |\Gamma(x)|)$ replacing $D^k(\log \Gamma(x))$ in (11), remain valid on this set. Formula (9) (VII, p. 307) shows that $\Gamma(x) \sim 1/x$ as x tends to 0, whence, by (7) of VII, p. 307, as x tends to $-n$ ($n$ an integer $\geq 0$). The function $1/\Gamma(x)$ can then be extended by continuity to all of $\mathbf{R}$, assigning it the value 0 at integers $\leq 0$; then, for all $x \in \mathbf{R}$ and and one shows as in prop. 2 of VII, p. 307, that the infinite product on the right of (13) is absolutely and uniformly convergent on every compact interval of $\mathbf{R}$.

Since $\Gamma(x) > 0$ for $x > 0$, equation (7) of VII, p. 307, shows that $\Gamma(x) < 0$ for $-(2n-1) < x < -(2n-2)$ and $\Gamma(x) > 0$ for

($n$ an integer $\geq 1$); also $\Gamma(x)$ has right limit $+\infty$ at the points $-2n$ and $-\infty$ at the points $-(2n+1)$, and has left limit $-\infty$ at the points $-2n$ and $+\infty$ at the points $-(2n+1)$ (for all $n \in \mathbf{N}$). Formula (11) of VII, p. 308, shows that, for $k = 2$, the right-hand side is always $\geq 0$ when it is defined, so

and consequently $\Gamma''(x)$ has the same sign as $\Gamma(x)$; thus $\Gamma$ is convex for $x > 0$ and for $-(2n+2) < x < -(2n+1)$, and concave for $-(2n+1) < x - 2n \ (n \in \mathbf{N})$; one deduces from this that, on the intervals where $\Gamma$ is convex, $\Gamma'(x)$ increases from $-\infty$ to $+\infty$, and on the intervals where $\Gamma$ is concave, $\Gamma'(x)$ decreases from $+\infty$ to $-\infty$. Whence the graph of $\Gamma$ (fig. 1).

Fig. 1