,"$undefined"]}]]}]}]}] 30:T1339,

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 $\int_0^\infty x^{s-1}f(x) dx = \Gamma(s) \int_0^\infty t^{-s}g(t) dt$.

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:
- $B(x, y) = B(y, x)$
- $B(x, y+1) = \frac{y}{x+y} B(x, y)$
- $B(x+1, y) = \frac{x}{x+y} B(x, y)$
- $B(x, y) = B(x+1, y) + B(x, y+1)$

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.

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 $x_1^k + \dots + x_n^k \le R^k$.

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