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The Beta-Gamma identity

Statement

For $Re p > 0$ and $Re q > 0$ ,

$B (p, q) = \frac{Γ (p) Γ (q)}{Γ (p + q)} .$

Facts & Assumptions

Given: Complex numbers $p, q$ with positive real parts.

[L1]

The real Beta-Gamma identity holds for positive real parameters (The real Beta--Gamma identity).

[L2]

On positive real arguments, the complex Gamma function agrees with the real Gamma function (The complex Gamma function restricts to the real Gamma function).

[L3]

If two holomorphic functions on a complex domain agree on a set with an accumulation point, then they agree everywhere (Identity theorem for holomorphic functions).

[L4]

Finite-interval parameter integrals of holomorphic kernels are holomorphic (A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic).

[L5]

Gamma is holomorphic on the right half-plane (Euler's Gamma function is holomorphic on the right half-plane).

[L6]

A locally uniform limit of holomorphic functions is holomorphic (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).

[L7]

Euler's Beta and Gamma functions are defined by their classical integrals (Euler's Beta function on the right half-planes, Euler's Gamma function on the right half-plane).

Proof

technique · direct

1.1givenL4L6L7

Fix a real number $q_{0} > 0$ . For $n \geq 2$ , define $B_{n} (p) : = \int_{1 / n}^{1 - 1 / n} t^{p - 1} (1 - t)^{q_{0} - 1} d t .$ By [L4], each $B_{n}$ is holomorphic on $Re p > 0$ . On a compact set $K \subseteq {p : Re p > 0}$ choose $a > 0$ with $Re p \geq a$ on $K$ ; then the omitted tails are dominated by $t^{a - 1} (1 - t)^{q_{0} - 1}$ near $0$ and by $(1 - t)^{q_{0} - 1}$ near $1$ , so $B_{n} \to B (\cdot, q_{0})$ locally uniformly on $Re p > 0$ . Hence [L6] makes $p \mapsto B (p, q_{0})$ holomorphic there.

2.1step 1.1L1L2L3L5

The function $H_{q_{0}} (p) : = Γ (p + q_{0}) B (p, q_{0}) - Γ (p) Γ (q_{0})$ is holomorphic on $Re p > 0$ by step 1.1 and [L5]. If $p > 0$ is real, then [L1] and [L2] identify the complex and real formulas, so $H_{q_{0}} (p) = 0$ . The positive real axis has an accumulation point in the right half-plane, so [L3] gives $H_{q_{0}} \equiv 0$ . Thus $Γ (p + q_{0}) B (p, q_{0}) = Γ (p) Γ (q_{0}) (Re p > 0, q_{0} > 0 real) .$

3.1step 2.1L3L5L6L7

Now fix $p$ with $Re p > 0$ . Repeating step 1.1 with the roles of $p$ and $q$ reversed shows that $q \mapsto B (p, q)$ is holomorphic on $Re q > 0$ . Therefore $K_{p} (q) : = Γ (p + q) B (p, q) - Γ (p) Γ (q)$ is holomorphic on $Re q > 0$ by [L5]. Step 2.1 shows $K_{p} (q) = 0$ for every positive real $q$ , so [L3] gives $K_{p} \equiv 0$ . Thus $Γ (p + q) B (p, q) = Γ (p) Γ (q)$ for all parameters in the stated half-planes.

4.1step 2.1step 3.1L3L5L7algebra∎

To justify division, suppose $Γ (z) = 0$ for some $Re z > 0$ . By [L7], $B (z, 1) = \int_{0}^{1} t^{z - 1} d t = 1 / z \neq 0$ , since $t^{z} \to 0$ as $t ↓ 0$ . Step 3.1 makes $q \mapsto B (z, q)$ holomorphic, hence continuous at $1$ , so $B (z, q) \neq 0$ for every real $q$ in some interval about $1$ contained in $(0, \infty)$ . Step 2.1 then gives $Γ (z + q) = 0$ throughout that interval. Its translated interval has an accumulation point in the right half-plane, so [L3] and [L5] force $Γ$ to vanish identically there, contradicting $Γ (1) = \int_{0}^{\infty} e^{- t} d t = 1$ from [L7]. Thus Gamma has no zeros in the right half-plane; in particular $Γ (p + q) \neq 0$ , and division of step 3.1 gives the stated identity.

Depends on

Used by

The standard intertwining operator A(nu) Definition
B(1/2,1/2)=π Example
K-type eigenvalues of A(nu): recurrence, closed form and nonvanishing Lemma
Euler's limit formula for Gamma Theorem

Cited to discharge well-definedness by Euler's Beta function on the right half-planes.

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1(vi) (standard reference, not scraped)
M. Weber, Complex Analysis, §3.7 (standard reference, not scraped)