Gamma function
= Gamma function
{title2=$\Gamma$}
{wiki}
The Gamma function extends <factorials> to a continuous argument. Its defining <Gamma integral> is
$$
\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt,\qquad \operatorname{Re}z>0,
$$
where $t^{z-1}=\exp((z-1)\log t)$ uses the real <natural logarithm> for $t>0$. The <Gamma function recurrence> gives its <meromorphic continuation> to the <complex plane>, with <simple poles> at $0,-1,-2,\ldots$ and no zeros. See <Gamma function residue at a nonpositive integer> and <Gamma function has no zeros> for proof outlines.https://dlmf.nist.gov/5.2[NIST DLMF, Section 5.2(i): definition and analytic properties]{ref}
For nonnegative <integers> $n$, the <factorial> identity has a shift of one:
$$
\Gamma(n+1)=n!,\qquad \Gamma(1)=1,\quad \Gamma(2)=1,\quad \Gamma(3)=2,\quad \Gamma(4)=6.
$$
Thus $\Gamma(n)=(n-1)!$ for positive <integers> $n$. Half-integer arguments supply a different familiar constant:
$$
\Gamma\left(\frac12\right)=\sqrt\pi,\qquad \Gamma\left(\frac32\right)=\frac{\sqrt\pi}{2}.
$$
The first value follows from the <Gaussian integral>; the second follows from the <Gamma function recurrence>.https://dlmf.nist.gov/5.4[NIST DLMF, equations 5.4.1 and 5.4.6: integer and half-integer values]{ref}
\Image[/gamma-function-factorial-interpolation.png]
{title=The Gamma function on the positive real axis passes through the factorial values 0! through 5! at arguments 1 through 6}
The plot shows only positive real arguments; the <gamma function> also accepts <complex numbers> away from its <poles>. Matching the <factorials> alone does not select a unique interpolation, even if the <Gamma function recurrence> is imposed: $f(x)=\Gamma(x)\exp(\sin(2\pi x))$ has the same positive-integer values and the same <recurrence relation>. The <Bohr-Mollerup theorem> supplies the extra condition that selects the <gamma function>.
The useful idea is to replace a discrete <product of numbers> by an <integral> whose exponent varies continuously. The <Gamma integral> then evaluates whole families of <integrals> at once, provides the <normalizing constant> for the <Gamma distribution> in <probability theory>, and appears in <physics>, for example in the <Bose integral>. Its large-positive-argument growth reconnects this continuous description to <factorials> through <Stirling's formula>.https://dlmf.nist.gov/5.11[NIST DLMF, equation 5.11.3: Gamma asymptotic expansion]{ref}
Historically, <Leonhard Euler> studied the interpolation of <factorials> in his paper E19, written in 1729 and published in 1738. His integral over $(0,1)$ becomes the <Gamma integral> after a <change of variables>, with the same one-step shift in the argument.https://scholarlycommons.pacific.edu/euler-works/19/[Euler Archive, E19: original-paper record, dates, and integral construction]{ref}