Multiple Integrals with Beta & Gamma

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Gamma Function Definition: $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} dt$, for $\text{Re}(z) > 0$. Properties: $\Gamma(z+1) = z\Gamma(z)$ For integer $n \ge 0$, $\Gamma(n+1) = n!$ $\Gamma(1) = 1$, $\Gamma(1/2) = \sqrt{\pi}$ Reflection Formula: $\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$ Duplication Formula (Legendre): $\Gamma(z)\Gamma(z+1/2) = 2^{1-2z}\sqrt{\pi}\Gamma(2z)$ Mellin Transform: If $f(x) = \int_0^\infty g(t)e^{-xt} dt$, then $\int_0^\infty x^{s-1}f(x) dx = \Gamma(s) \int_0^\infty t^{-s}g(t) dt$. Beta Function Definition: $B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt$, for $\text{Re}(x) > 0, \text{Re}(y) > 0$. Relation to Gamma Function: $B(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$. Alternative Forms: $B(x, y) = 2\int_0^{\pi/2} \sin^{2x-1}(\theta)\cos^{2y-1}(\theta) d\theta$ $B(x, y) = \int_0^\infty \frac{t^{x-1}}{(1+t)^{x+y}} dt$ Properties: $B(x, y) = B(y, x)$ $B(x, y+1) = \frac{y}{x+y} B(x, y)$ $B(x+1, y) = \frac{x}{x+y} B(x, y)$ $B(x, y) = B(x+1, y) + B(x, y+1)$ Dirichlet Integral Generalized Dirichlet Integral: $$ \int \dots \int_R x_1^{a_1-1} \dots x_n^{a_n-1} dx_1 \dots dx_n = \frac{\Gamma(a_1)\dots\Gamma(a_n)}{\Gamma(a_1+\dots+a_n+1)} $$ where $R$ is the region $x_i \ge 0$ and $x_1+\dots+x_n \le 1$. Extension: $$ \int \dots \int_R f(x_1+\dots+x_n) x_1^{a_1-1} \dots x_n^{a_n-1} dx_1 \dots dx_n = \frac{\Gamma(a_1)\dots\Gamma(a_n)}{\Gamma(a_1+\dots+a_n)} \int_0^1 f(u) u^{a_1+\dots+a_n-1} du $$ for $R$ as above. Multiple Integrals and Transformations Change of Variables (Jacobian): $$ \iint_R f(x,y) dx dy = \iint_S f(x(u,v), y(u,v)) \left| \frac{\partial(x,y)}{\partial(u,v)} \right| du dv $$ where $\frac{\partial(x,y)}{\partial(u,v)} = \det \begin{pmatrix} \partial x / \partial u & \partial x / \partial v \\ \partial y / \partial u & \partial y / \partial v \end{pmatrix}$. Spherical Coordinates (3D): $x = \rho \sin\phi \cos\theta$ $y = \rho \sin\phi \sin\theta$ $z = \rho \cos\phi$ Jacobian: $J = \rho^2 \sin\phi$ $dV = \rho^2 \sin\phi d\rho d\phi d\theta$ Generalized Spherical Coordinates (n-D): $$ \int_0^\infty \dots \int_0^\infty f(x_1^k + \dots + x_n^k) x_1^{a_1-1} \dots x_n^{a_n-1} dx_1 \dots dx_n $$ $$ = \frac{1}{k^n} \frac{\Gamma(a_1/k)\dots\Gamma(a_n/k)}{\Gamma((a_1+\dots+a_n)/k)} \int_0^\infty f(u) u^{(a_1+\dots+a_n)/k - 1} du $$ This transformation is useful for integrals over regions like $x_1^k + \dots + x_n^k \le R^k$. Applications & Examples Volume of n-Sphere: The volume of an n-dimensional sphere of radius $R$ is $V_n(R) = \frac{\pi^{n/2} R^n}{\Gamma(n/2+1)}$. Gaussian Integral Generalization: $\int_{-\infty}^\infty e^{-ax^2} dx = \sqrt{\frac{\pi}{a}}$. $$ \int_{-\infty}^\infty \dots \int_{-\infty}^\infty e^{-(x_1^2+\dots+x_n^2)} dx_1 \dots dx_n = \pi^{n/2} $$ Beta Integral for Area/Volume: Consider integral $\iint_R x^{a-1}y^{b-1} dx dy$ over $x+y \le 1, x,y \ge 0$. This is $B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Example Integral: $$ \int_0^1 \int_0^{1-x} \sqrt{xy(1-x-y)} dy dx $$ Let $x=u$, $y=v(1-u)$. Jacobian is $1-u$. The integral becomes $$ \int_0^1 \int_0^1 \sqrt{u \cdot v(1-u) \cdot (1-u-v(1-u))} (1-u) dv du $$ $$ = \int_0^1 \int_0^1 \sqrt{u v (1-u)^2 (1-v)} (1-u) dv du $$ $$ = \int_0^1 u^{1/2} (1-u)^2 du \int_0^1 v^{1/2} (1-v)^{1/2} dv $$ $$ = B(3/2, 3) \cdot B(3/2, 3/2) $$ $$ = \frac{\Gamma(3/2)\Gamma(3)}{\Gamma(9/2)} \cdot \frac{\Gamma(3/2)\Gamma(3/2)}{\Gamma(3)} $$ $$ = \frac{(\sqrt{\pi}/2) \cdot 2!}{(7!! \sqrt{\pi}/8)} \cdot \frac{(\sqrt{\pi}/2)(\sqrt{\pi}/2)}{2!} = \frac{\pi}{48} $$