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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

Dependency tree · 2 levels

Sources

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §2.1