Frullani integral explained

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

\int { ^{inf ty} _{0} (f(ax) - f(bx))/(x) }\,xwhere f is a function defined for all non-negative real numbers that has a limit at inf ty , which we denote by f( inf ty) .

The following formula for their general solution holds if

f is continuous on ( 0 , inf ty) , has finite limit at inf ty , and a,b > 0 :\int { ^{inf ty} _{0} (f(ax) - f(bx))/(x) }\,x=\Big(f(\infty)-f(0)\Big)\ln .

Proof for continuously differentiable functions

A simple proof of the formula (under stronger assumptions than those stated above, namely

f \in l{C} ^{1(0,infty)} ) can be arrived at by using the Fundamental theorem of calculus to express the integrand as an integral of f'(xt) = (\partial)/(\left( \partial t) (f(xt))/(x) \right) :\begin{align} (f(ax) - f(bx))/(x) & = \left[ (f(xt))/(x) \right] ^{t = a} _{t = b} \\ & = \int ^{a} _{b} f'(xt) dt \\ \end{align}

and then use Tonelli’s theorem to interchange the two integrals:

\begin{align} \int ^{inf ty} _{0} (f(ax) - f(bx))/(x) dx & = \int ^{inf ty} _{0} \int ^{a} _{b} f'(xt) dt dx \\ & = \int ^{a} _{b} \int ^{inf ty} _{0} f'(xt) dx dt \\ & = \int ^{a} _{b} \left[ (f(xt))/(t) \right] ^{x \to inf ty} _{x =0} dt \\ & = \int ^{a} _{b} (f( inf ty) - f( 0 ))/(t) dt \\ & = ( f( inf ty) - f( 0 ) ) ( ln (a) -ln (b) ) \\ & = ( f( inf ty) - f( 0 ) ) ln ( (a)/(b) ) \\ \end{align} Note that the integral in the second line above has been taken over the interval [b,a] , not [a,b] .

Applications

ln (x) by letting f(x) = e ^{-x} and a =1 :{\int { ^{inf ty} _{0} (e ^{-x} - e ^{-bx})/(x) }\,x=\Big(\lim_\frac-e^0\Big)\ln \Big(}\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.

Notes and References

1. Bravo . Sergio . Gonzalez . Ivan . Kohl . Karen . Moll . Victor Hugo . Victor Hugo Moll . Integrals of Frullani type and the method of brackets . Open Mathematics . 21 January 2017 . 15 . 1 . 10.1515/math-2017-0001 . 17 June 2020. free .