Frullani's integral

Giuliano Frullani

Quick Info

Born: 1795, Italian.

Died:  1834

Introduction

The Italian mathematician G. Frullani, 1795-1834, reported to G. A. Plana, 1781-1864, the formula

\begin{equation}  \int_{0}^{\infty }\frac{f\left ( ax \right )-f\left ( bx \right )}{x}dx=f\left ( 0 \right )log\left ( \frac{a}{b} \right )\end{equation}

In 1823 and 1827, Cauchy gave a satisfactory proof of the formula 

\begin{equation} \int_{0}^{\infty }\frac{f\left ( ax \right )-f\left ( bx \right )}{x}dx=\left [ f\left ( \infty  \right )-f\left ( 0 \right ) \right ]log\left ( \frac{a}{b} \right ) \end{equation}

We look at a nice Integral identity that alows us to evaluate an entire family of integrals.

which is called Frullani’s integral, after the Italian mathematician Giuliano Frullani (1795–1834).

For example:

Solution:

References

1  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