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A rational-kernel Frullani integral

Example

For $a, b > 0$ , $\int_{0}^{\infty} \frac{(1 + a x)^{- 1} - (1 + b x)^{- 1}}{x} d x = \int_{a}^{b} \frac{d t}{t} .$

Facts & Assumptions

Given: Positive $a, b$ and $f (t) = 1 / (1 + t)$ .

[L1]

Frullani's formula gives $(f (0) - L) \int_{a}^{b} d t / t$ when $f (t) \to L$ (Frullani's formula with its proper integral factor).

[L2]

The $p$ -test gives convergence of $\int_{1}^{\infty} x^{- 2} d x$ (The improper $p$ -test for rational exponents).

Verification

technique · computation

1.1

Here $f$ is continuous and $f (0) = 1$ . Also $0 < f (t) = 1 / (1 + t) < ε$ whenever $t > 1 / ε$ , so $f (t) \to 0$ directly from the definition of the limit at infinity. Thus [L1] gives exactly the displayed identity.

1.2

For $x > 0$ , the integrand simplifies to [L2] $\frac{b - a}{(1 + a x) (1 + b x)} .$ It has finite limit $b - a$ at zero and is bounded in absolute value by a constant multiple of $x^{- 2}$ for $x \geq 1$ . This independently confirms local convergence at zero and tail convergence by [L2], without replacing the proper factor by a logarithm. ∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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

William F. Trench, Introduction to Real Analysis, Frullani integral exercise (standard reference, not scraped)