Английская Википедия:Frullani integral

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

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

where <math>f</math> is a function defined for all non-negative real numbers that has a limit at <math>\infty</math>, which we denote by <math>f(\infty)</math>.

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

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

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 <math>f'(xt) = \frac{\partial }{\partial t} \left(\frac{f(xt)}{x}\right)</math>:

<math>\begin{align}

 \frac{f(ax)-f(bx)}{x} &= \left[\frac{f(xt)}{x}\right]_{t=b}^{t=a} \, \\
  & = \int_b^a f'(xt) \, dt \\
\end{align}</math>

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

<math>\begin{align}

 \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{align}</math>

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

Applications

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

<math>{\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)</math>

The formula can also be generalized in several different ways.

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.