In mathematics, Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. The integrals are of the form
\( {\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x} \)
where f f is a function defined for all non-negative real numbers that has a limit at ∞ \infty , which we denote by \( {\displaystyle f(\infty )} \) .
The following formula for their general solution holds under certain conditions:
\( {\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x={\Big (}f(\infty )-f(0){\Big )}\ln {\frac {a}{b}}.} \)
Proof
A simple proof of the formula can be arrived at by using the Fundamental theorem of calculus to express the integrand as an integral of \( {\displaystyle f'(xt)={\frac {\partial }{\partial t}}\left({\frac {f(xt)}{x}}\right)} \) :
\( {\displaystyle {\begin{aligned}{\frac {f(ax)-f(bx)}{x}}&=\left[{\frac {f(xt)}{x}}\right]_{t=b}^{t=a}\,\\&=\int _{b}^{a}f'(xt)\,dt\\\end{aligned}}} \)
and then use Tonelli’s theorem to interchange the two integrals:
\( {\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,dx&=\int _{0}^{\infty }\int _{b}^{a}f'(xt)\,dt\,dx\\&=\int _{b}^{a}\int _{0}^{\infty }f'(xt)\,dx\,dt\\&=\int _{b}^{a}\left[{\frac {f(xt)}{t}}\right]_{x=0}^{x\to \infty }\,dt\\&=\int _{b}^{a}{\frac {f(\infty )-f(0)}{t}}\,dt\\&={\Big (}f(\infty )-f(0){\Big )}{\Big (}\ln(a)-\ln(b){\Big )}\\&={\Big (}f(\infty )-f(0){\Big )}\ln {\Big (}{\frac {a}{b}}{\Big )}\\\end{aligned}}} \)
Note that the integral in the second line above has been taken over the interval \( {\displaystyle [b,a]} \), not \( [a,b] \).
Applications
The formula can be used to derive an integral representation for the natural logarithm \( \ln(x) \) by letting \( {\displaystyle f(x)=e^{-x}} \) and \( a=1 \) :
\( {\displaystyle {\int _{0}^{\infty }{\frac {e^{-x}-e^{-bx}}{x}}\,{\rm {d}}x={\Big (}\lim _{n\to \infty }{\frac {1}{e^{n}}}-e^{0}{\Big )}\ln {\Big (}{\frac {1}{b}}}{\Big )}=\ln(b)} \)
The formula can also be generalized in several different ways.[1]
References
G. Boros, Victor Hugo 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 Hugo (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.
vte
Lists of integrals
Rational functions Irrational functions Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Exponential functions Logarithmic functions Gaussian functions Definite integrals
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
