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Euler's Beta integral converges exactly for two positive parameters

Statement

Let $p, q$ be real. The Beta integral converges if and only if $p > 0$ and $q > 0$ .

Here the Beta integral means $\int_{0}^{1} t^{p - 1} (1 - t)^{q - 1} d t$ .

Facts & Assumptions

Given: Real parameters $p, q$ , with the integral split at $1 / 2$ .

[F1]

If $0 \leq u \leq v$ eventually at a finite singular endpoint and the improper integral of $v$ converges there, then the integral of $u$ converges (Comparison tests for improper integrals).

[F2]

For $x > 0$ , $\log^{'} (x) = 1 / x$ and $\log x = \int_{1}^{x} d t / t$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t).

[F3]

The natural logarithm is strictly increasing and maps $(0, \infty)$ onto $R$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

[F4]

For real $α$ , $x^{α}$ is differentiable on $(0, \infty)$ with derivative $α x^{α - 1}$ (Continuity and derivatives of positive-base real powers).

Proof

technique · direct

1.1givenF1F2F3F4algebra

On $(0, 1 / 2]$ , the continuous positive factor $(1 - t)^{q - 1}$ is bounded above and below by positive constants. Thus [F1] reduces convergence at zero to that of $\int_{0}^{1 / 2} t^{p - 1} d t$ , whose primitive from [F4] converges exactly for $p > 0$ ; at $p = 0$ , [F2] and [F3] give logarithmic divergence, and for $p < 0$ the integrand dominates the same threshold.

1.2givenF1F2F3F4algebra

The substitution $u = 1 - t$ changes the end $t ↑ 1$ into $u ↓ 0$ . There $t^{p - 1}$ is bounded above and below by positive constants, so [F4] and the same comparison give convergence exactly for $q > 0$ , with logarithmic divergence at $q = 0$ .

2.1step 1.1step 1.2algebra∎

Both endpoint integrals converge exactly when $p > 0$ and $q > 0$ . If either parameter is nonpositive, the corresponding endpoint diverges by step 1.1 or step 1.2.

Depends on

Used by

Symmetry and the trigonometric form of the real Beta integral Proposition
The real Beta--Gamma identity Theorem

Cited to discharge well-definedness by Euler's real Beta integral.

Dependency tree · two levels

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Sources

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §2.1 (standard reference, not scraped)