Gamma函数和Beta函数

A. Gamma Function

For $s > 0$ define \begin{equation}\label{1} \Gamma(s) = \int_0^{\infty}t^{s - 1}e^{-t} \,{\rm d}t. \end{equation}

$\Gamma$ is called the gamma function. Moreover, the gamma function extends to an holomorphic function in the half-plane $\{z \in \mathbb{C}\colon {\rm Re} \ z > 0\}$, and is still given there by the integral formula \eqref{1}. Furthermore, by Taylor expansion it has a meromorphic extension on the whole plane and has simple poles $z_j = -j, j = 0, 1, 2, \cdots$ with residues $(-1)^j/j!$.

Here are some properties of the gamma function:

1. $\Gamma(z + 1) = z\Gamma(z), \Gamma(n) = (n - 1)!, \Gamma(1/2) = \sqrt{\pi}$;
2. (Reflection formula) $$\Gamma(z)\Gamma(1 - z) = \frac{\pi}{\sin\pi z};$$
3. (Duplication formula) $$\Gamma(2z) = \frac{2^{2z - 1}}{\sqrt{\pi}}\Gamma(z)\Gamma(z + 1/2);$$
4. (Stirling's formula) $$\lim\limits_{x \rightarrow \infty}\frac{\Gamma(x + 1)}{(x/e)^x\sqrt{2\pi x}} = 1.$$ Consequently, for any $t > 0$ we have $$\lim\limits_{x \rightarrow \infty}\frac{\Gamma(x)}{\Gamma(x + t)} = 0;$$
5. (Euler's limit formula) For $n \in \mathbb{Z}_+$ and ${\rm Re} \ z > 0$ define $$\Gamma_n(z) = \int_0^n\left(1 - \frac{t}{n}\right)^nt^{z - 1} \,{\rm d}t.$$ Then $$\Gamma_n(z) = \frac{n!n^z}{z(z + 1) \cdots (z + n)}$$ and $\lim\limits_{n \rightarrow \infty}\Gamma_n(z) = \Gamma(z)$;
6. (Infinite product form of Euler's limit formula) $$\frac{1}{\Gamma(z)} = ze^{\gamma z}\prod\limits_{k = 1}^{\infty}\left(1 + \frac{z}{k}\right)e^{-z/k},$$ where ${\rm Re} \ z > 0$ and $\gamma$ is Euler's constant: $$\gamma = \lim\limits_{n \rightarrow \infty}\left(\sum\limits_{k = 1}^n\frac{1}{k} - \ln n\right);$$
7. For $x$ and $y$ real with $x \notin \{0, -1, -2, \cdots \}$, we have $$\frac{1}{|\Gamma(x + iy)|^2} = \frac{1}{|\Gamma(x)|^2}\prod\limits_{k = 0}^{\infty}\left(1 + \frac{y^2}{(k + x)^2}\right).$$ Hence $|\Gamma(x + iy)| \leq |\Gamma(x)|$ and also that $$\frac{1}{|\Gamma(x + iy)|^2} \leq \frac{1}{|\Gamma(x)|^2}e^{C(x)|y|^2},$$ where $$C(x) = \frac{1}{2}\sum\limits_{k = 0}^{\infty}\frac{1}{(k + x)^2},$$ whenever $x \in \mathbb{R} \smallsetminus \{0, -1, -2, \cdots\}$ and $y \in \mathbb{R}$. We also have a simpler expression for this estimate when $x > 0$ and $y \in \mathbb{R}$: $$\frac{1}{|\Gamma(x + iy)|} \leq \frac{1}{|\Gamma(x)|}e^{\max\{x^{-2}, x^{-1}\}|y|^2}.$$ For $x = -N \in \{0, -1, -2, \cdots\}$ and $y \in \mathbb{R}$ we have $$\frac{1}{|\Gamma(-N + iy)|^2} \leq |iy||1 + iy||2 + iy| \cdots |N + iy|e^{|y|^2}.$$

B. Beta Function

The beta function is defined as \begin{equation}\label{2} B(p, q) = \int_0^1t^{p - 1}(1 - t)^{q - 1} \,{\rm d}t, \end{equation} where $p, q > 0$. Analogously, the beta function also extends to an holomorphic function in $\{(z, w) \in \mathbb{C} \times \mathbb{C}\colon {\rm Re} \ z > 0, {\rm Re} \ w > 0\}$ which is still given by \eqref{2} and has a meromorphic extension on $\mathbb{C} \times \mathbb{C}$. Clearly $B(z, w) = B(w, z)$. We also have the following relationship between the gamma and the beta function: $$B(z, w) = \frac{\Gamma(z)\Gamma(w)}{\Gamma(z + w)}.$$

C. Applications

C.1 Integral Formulas

1. Let $k_1, \cdots, k_n$ be nonngeative even integers. Note that the integral $$\int_{\mathbb{R}^n}x_1^{k_1} \cdots x_n^{k_n}e^{-|x|^2} \,{\rm d}x = \prod\limits_{j = 1}^n\int_{-\infty}^{+\infty}x_j^{k_j}e^{-x_j^2} \,{\rm d}x_j = \prod\limits_{j = 1}^n\Gamma\left(\frac{k_j + 1}{2}\right)$$ expressed in polar coordinates is equal to $$\left(\int_{\mathbb{S}^{n - 1}}\theta_1^{k_1} \cdots \theta_n^{k_n} \,{\rm d}\theta\right)\int_0^{\infty}r^{k_1 + \cdots + k_n}r^{n - 1}e^{-r^2} \,{\rm d}r.$$ It follows that $$\int_{\mathbb{S}^{n - 1}}\theta_1^{k_1} \cdots \theta_n^{k_n} \,{\rm d}\theta = 2\Gamma\left(\frac{k_1 + \cdots + k_n + n}{2}\right)^{-1}\prod\limits_{j = 1}^n\Gamma\left(\frac{k_j + 1}{2}\right);$$
   
2. Using the change of variables $u = \sin^2\varphi$ in the definition of the beta function, we have the following classical result: $$\int_0^{\pi/2}\sin^a\varphi\cos^b\varphi \,{\rm d}\varphi = \frac{1}{2}B\left(\frac{a + 1}{2}, \frac{b + 1}{2}\right) = \frac{1}{2}\frac{\Gamma\left(\frac{a + 1}{2}\right)\Gamma\left(\frac{b + 1}{2}\right)}{\Gamma\left(\frac{a + b}{2} + 1\right)},$$ where ${\rm Re} \ a, {\rm Re} \ b > -1$.

C.2 Voulme of the Unit Ball and Surfaces of the Unit Sphere

We denote by $v_n$ the volume of the unit ball in $\mathbb{R}^n$ and by $w_{n - 1}$ the surface area of the unit sphere $\mathbb{S}^{n - 1}$. We have the following: $$v_n = \frac{\pi^{n/2}}{\Gamma(n/2 + 1)} = \frac{2\pi^{n/2}}{n\Gamma(n/2)}$$ and $$w_{n - 1} = nv_n = \frac{2\pi^{n/2}}{\Gamma(n/2)}.$$

 

Ref: Grafakos, L. Classical Fourier Analysis.