Type of improper integral with general solution
In mathematics, Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. The integrals are of the form
![{\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1990c93a26b6a1c3f068af6623d6af1f6e15da5)
where
is a function defined for all non-negative real numbers that has a limit at
, which we denote by
.
The following formula for their general solution holds if
is continuous on
, has finite limit at
, and
:
![{\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x={\Big (}f(\infty )-f(0){\Big )}\ln {\frac {a}{b}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c6350bdb6036ee225fc0868117ae5cc4ef5e08)
Proof for continuously differentiable functions
A simple proof of the formula (under stronger assumptions than those stated above, namely
) can be arrived at by using the Fundamental theorem of calculus to express the integrand as an integral of
:
![{\displaystyle {\begin{aligned}{\frac {f(ax)-f(bx)}{x}}&=\left[{\frac {f(xt)}{x}}\right]_{t=b}^{t=a}\,\\&=\int _{b}^{a}f'(xt)\,dt\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c88324e51ed55a11b3a5c11fd54d29f1cf3d6f3)
and then use Tonelli’s theorem to interchange the two integrals:
![{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,dx&=\int _{0}^{\infty }\int _{b}^{a}f'(xt)\,dt\,dx\\&=\int _{b}^{a}\int _{0}^{\infty }f'(xt)\,dx\,dt\\&=\int _{b}^{a}\left[{\frac {f(xt)}{t}}\right]_{x=0}^{x\to \infty }\,dt\\&=\int _{b}^{a}{\frac {f(\infty )-f(0)}{t}}\,dt\\&={\Big (}f(\infty )-f(0){\Big )}{\Big (}\ln(a)-\ln(b){\Big )}\\&={\Big (}f(\infty )-f(0){\Big )}\ln {\Big (}{\frac {a}{b}}{\Big )}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf1102ef7fa80c28df05770e45f503ac74fa0714)
Note that the integral in the second line above has been taken over the interval
, not
.
Applications
The formula can be used to derive an integral representation for the natural logarithm
by letting
and
:
![{\displaystyle {\int _{0}^{\infty }{\frac {e^{-x}-e^{-bx}}{x}}\,{\rm {d}}x={\Big (}\lim _{n\to \infty }{\frac {1}{e^{n}}}-e^{0}{\Big )}\ln {\Big (}{\frac {1}{b}}}{\Big )}=\ln(b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce9930254c9571dc3e65dac3fe625115c8aace8)
The formula can also be generalized in several different ways.[1]
References