Special Functions: Euler's Gamma & Beta Functions

1. Definition of the Gamma Function

The Euler Gamma Function $\Gamma(z)$ is defined for all complex numbers with positive real part $\operatorname{Re}(z) > 0$ (and for all real $x > 0$) by the improper integral:

2. The Fundamental Recurrence Relation: $\Gamma(z+1) = z\Gamma(z)$

Applying integration by parts with $u = t^z \implies du = z t^{z-1} dt$ and $dv = e^{-t} dt \implies v = -e^{-t}$:

For $\operatorname{Re}(z) > 0$, $\lim_{t \to \infty} t^z e^{-t} = 0$ and $\lim_{t \to 0^+} t^z e^{-t} = 0$. Thus:

3. Factorial Interpolation for Non-Negative Integers

Evaluating the base case at $z = 1$:

By mathematical induction, for any positive integer $n \in \mathbb{N}$:

Thus, $\Gamma(z)$ extends the discrete factorial function continuously across the real continuum: $\Gamma(n) = (n - 1)!$.

1. The Fundamental Half-Integer Value: $\Gamma(1/2) = \sqrt{\pi}$

Setting $z = 1/2$ in the definition:

Substitute $t = u^2 \implies dt = 2u \, du$:

Using the recurrence $\Gamma(x+1) = x\Gamma(x)$, higher half-integers are directly calculated:

2. Euler's Reflection Formula

For any non-integer $z \in \mathbb{C} \setminus \mathbb{Z}$:

This remarkable identity links Gamma functions to circular trigonometry. Notice that for $z = 1/2$, $\Gamma(1/2)^2 = \frac{\pi}{\sin(\pi/2)} = \pi \implies \Gamma(1/2) = \sqrt{\pi}$.

3. Legendre's Duplication Formula

1. Definition of the Beta Function

The Euler Beta Function $B(p, q)$ is defined for $p, q > 0$ by the compact interval integral:

2. Fundamental Symmetry: $B(p, q) = B(q, p)$

Substituting $u = 1 - t \implies dt = -du$ swaps the endpoints $t = 0 \to u = 1$ and $t = 1 \to u = 0$:

3. Alternative Representations

• Trigonometric Form: Substitute $t = \sin^2\theta \implies dt = 2\sin\theta\cos\theta \, d\theta$: $$\mathbf{B(p, q) = 2 \int_0^{\pi/2} \sin^{2p-1}(\theta) \cos^{2q-1}(\theta) \, d\theta}$$
• Infinite Domain Form: Substitute $t = \frac{u}{1 + u} \implies 1 - t = \frac{1}{1 + u}, dt = \frac{du}{(1+u)^2}$: $$\mathbf{B(p, q) = \int_0^\infty \frac{u^{p-1}}{(1 + u)^{p + q}} \, du}$$

1. The Bridge Theorem

2. Rigorous Multi-Dimensional Transformation Proof

Consider the product of two Gamma functions expressed under substitutions $t = x^2$ and $s = y^2$:

Multiplying these two independent single integrals produces a double integral over the first quadrant $Q_1 = \{ (x, y) : x \ge 0, y \ge 0 \}$ of $\mathbb{R}^2$:

Transform into 2D polar coordinates: $x = r\cos\theta, y = r\sin\theta, dx\,dy = r\,dr\,d\theta$ with $r \in [0, \infty)$ and $\theta \in [0, \pi/2]$:

Recognize both parenthetical factors:

1. The radial factor: substituting $u = r^2 \implies du = 2r\,dr$ gives $2 \int_0^\infty r^{2(p+q)-1} e^{-r^2} dr = \int_0^\infty u^{(p+q)-1} e^{-u} du = \mathbf{\Gamma(p + q)}$.
2. The angular factor: by the trigonometric form of the Beta function from Section 6.3: $$2 \int_0^{\pi/2} \sin^{2q-1}(\theta) \cos^{2p-1}(\theta) \, d\theta = \mathbf{B(p, q)}$$

Therefore:

1. Volume of the $n$-Dimensional Euclidean Ball $V_n(R)$

Let $B_n(R) = \{ x \in \mathbb{R}^n : \sum_{i=1}^n x_i^2 \le R^2 \}$ be the $n$-dimensional Euclidean ball of radius $R$. Its volume scales as $V_n(R) = C_n R^n$.

Integrating the multi-variable Gaussian over $\mathbb{R}^n$:

Converting to spherical shell integration $d^n x = S_{n-1}(r) dr = n C_n r^{n-1} dr$:

With substitution $u = r^2$, the integral evaluates to $\frac{1}{2}\Gamma(n/2)$. Thus:

2. Verification in Low Dimensions

• $n = 1$: $V_1(R) = \frac{\pi^{1/2}}{\Gamma(3/2)} R = \frac{\sqrt{\pi}}{\frac{1}{2}\sqrt{\pi}} R = 2R$ (Line segment length)
• $n = 2$: $V_2(R) = \frac{\pi^1}{\Gamma(2)} R^2 = \pi R^2$ (Circular disk area)
• $n = 3$: $V_3(R) = \frac{\pi^{3/2}}{\Gamma(5/2)} R^3 = \frac{\pi^{3/2}}{\frac{3}{4}\sqrt{\pi}} R^3 = \frac{4}{3}\pi R^3$ (3D sphere volume)
• $n = 4$: $V_4(R) = \frac{\pi^2}{\Gamma(3)} R^4 = \frac{\pi^2}{2} R^4$ (4D hypersphere volume)