The Gamma Function

The factorial $n!$ extends to a meromorphic function on the whole plane — the Gamma function $\Gamma (z)$ — and it is the first nontrivial special function, a showcase for every technique in the folder: an integral representation, a Weierstrass product, analytic continuation, and Stirling asymptotics from steepest descent. This page builds on infinite-products.md and analytic-continuation.md.

Three definitions

$\Gamma$ admits three equivalent definitions, each illuminating a different property:

• Euler integral (convergent for $\mathrm{Re}z>0$): $$\Gamma (z)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt.$$
• Euler limit (valid wherever $\Gamma$ is defined): $$\Gamma (z)=\underset{n\to \infty }{\lim }\frac{n!n^z}{z(z+1)\cdots (z+n)}.$$
• Weierstrass product (an entire function of $1/\Gamma$): $$\frac{1}{\Gamma (z)}=z{e}^{\gamma z}\prod _{n=1}^{\infty }(1+\frac{z}{n})e^{-z/n},$$ with $\gamma$ the Euler–Mascheroni constant. This is exactly a Hadamard factorization: $1/\Gamma$ is entire of order $1$, with simple zeros at $z=0,-1,-2,\dots$.

The functional equation and continuation

Integration by parts on the Euler integral gives the functional equation $$\Gamma (z+1)=z\Gamma (z),\Gamma (1)=1\Rightarrow \Gamma (n+1)=n!.$$ Rewritten as $\Gamma (z)=\Gamma (z+1)/z$, it analytically continues $\Gamma$ from the right half-plane leftward one strip at a time:

Poles of $\Gamma$. $\Gamma$ is meromorphic on $\mathbb{C}$ with simple poles at $z=0,-1,-2,\dots$ and no zeros; the residue at $z=-n$ is $\frac{(-1{)}^n}{n!}$.

This is the continuation glimpsed in analytic-continuation.md and the backbone of dimensional regularization in QFT, where the poles of $\Gamma (2-d/2)$ locate the ultraviolet divergences of loop integrals — a forward link collected in physics-bridge.md.

The reflection and duplication formulas

Pairing the Weierstrass product for $\Gamma (z)$ and $\Gamma (-z)$ against the product for $\sin$ gives the reflection formula $$\Gamma (z)\Gamma (1-z)=\frac{\pi }{\sin \pi z},$$ which instantly yields $\Gamma (\frac{1}{2})=\sqrt{\pi }$ (hence the Gaussian $\int e^{-t^2}dt=\sqrt{\pi }$) and exhibits the poles as mirror images of the zeros of $\sin$. The Legendre duplication formula $$\Gamma (z)\Gamma (z+\frac{1}{2})=2^{1-2z}\sqrt{\pi }\Gamma (2z)$$ is the $m=2$ case of the Gauss multiplication formula. Both are rigidity statements forced by the identity theorem: proved for real $z$, they hold on all of $\mathbb{C}$.

Stirling's asymptotics

The large-$|z|$ behaviour comes from applying steepest descent to the Euler integral (the integrand $t^z{e}^{-t}$ peaks at the saddle $t=z$):

Stirling's formula. As $|z|\to \infty$ in $|\arg z|<\pi$, $$\log \Gamma (z)=(z-\frac{1}{2})\log z-z+\frac{1}{2}\log 2\pi >+\frac{1}{12z}-\cdots ,$$ so $\Gamma (z)\sim \sqrt{2\pi }{z}^{z-1/2}{e}^{-z}$ and $n!\sim \sqrt{2\pi n}>(n/e{)}^n$.

The series is asymptotic, not convergent — the generic outcome of the saddle-point method (saddle-point.md).

Related functions

• The Beta function $B(x,y)={\int }_0^1{t}^{x-1}(1-t{)}^{y-1}dt=\Gamma (x)\Gamma (y)/\Gamma (x+y)$ — the multiplicative bridge between $\Gamma$-values.
• The digamma $\psi ={\Gamma }^{\prime }/\Gamma$ and its series $\psi (z)=-\gamma +{\sum }_{n\ge 0}(\frac{1}{n+1}-\frac{1}{n+z})$, a Mittag-Leffler expansion.
• $\Gamma$ enters the functional equation of the Riemann zeta function as the "archimedean factor" completing the Euler product.

References

• Ahlfors, Complex Analysis, Ch. 5 §2.4.
• Stein & Shakarchi, Complex Analysis, Ch. 6.
• Whittaker & Watson, A Course of Modern Analysis, Ch. XII.