Frullani integral

Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. The integrals are of the form

\int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x

where ${f(x)}$ is a function over ${x\geq 0}$ , and the limit of ${f(x)}$ exists at ${\infty }$ .

The following formula for their general solution holds under certain conditions:

{\int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x={\Big (}f(\infty )-f(0){\Big )}\ln {\frac {a}{b}}}

Proof

A simple proof of the formula can be arrived at by expanding the integrand into an integral, and then using Fubini's theorem to interchange the two integrals:

{\begin{aligned}\int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,dx&=\int _{0}^{\infty }\left[{\frac {f(xt)}{x}}\right]_{t=b}^{t=a}\,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=\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}}

Note that the integral in the second line above has been taken over the interval $[b,a]$ , not $[a,b]$ .

Applications

The formula can be used to derive an integral representation for the natural logarithm $\ln(x)$ by letting $f(x)=e^{-x}$ and $a=1$ :

{\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)

The formula can also be generalized in several different ways.[1]

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References

G. Boros, V. Moll, Irresistible Integrals (2004), pp. 98
Juan Arias-de-Reyna, On the Theorem of Frullani (PDF; 884 kB), Proc. A.M.S. 109 (1990), 165-175.
ProofWiki, proof of Frullani's integral.

Bravo, Sergio; Gonzalez, Ivan; Kohl, Karen; Moll, Victor H. (21 January 2017). "Integrals of Frullani type and the method of brackets". Open Mathematics. 15 (1). doi:10.1515/math-2017-0001. Retrieved 17 June 2020.

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[1] Bravo, Sergio; Gonzalez, Ivan; Kohl, Karen; Moll, Victor H. (21 January 2017). "Integrals of Frullani type and the method of brackets". Open Mathematics. 15 (1). doi:10.1515/math-2017-0001. Retrieved 17 June 2020.