Frullan integrals

As Frullan integrals , improper integrals become of the type

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

designated. They were first mentioned in a letter by Giuliano Frullani in 1821 and published in 1828. The following sentence applies:

Let be a for continuous function with and the finite limit value , then holds ${\displaystyle f(x)}$ ${\displaystyle x\geq 0}$ ${\displaystyle A:=f(0)}$ ${\displaystyle B:=\lim _{x\to \infty }f(x)}$

${\displaystyle \int \limits _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x=(A-B)\ln {\frac {b}{a}}}$.

Important examples arise for or with : ${\displaystyle f(x)=e^{-x}\ }$ ${\displaystyle f(x)=\arctan x\ }$ ${\displaystyle a,b>0}$

${\displaystyle \int \limits _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}}{x}}\,{\rm {d}}x=\ln {\frac {b}{a}}}$

${\displaystyle \int \limits _{0}^{\infty }{\frac {\arctan(ax)-\arctan(bx)}{x}}\,{\rm {d}}x={\frac {\pi }{2}}\ln {\frac {a}{b}}}$

literature

• G. Frullani , Sopra Gli Integrali Definiti, Memorie della Società Italiana delle Scienze, Modena, XX (1828), pp. 448-467.
• TJ Bromwich , An Introduction to the Theory of Infinite Series, Macmillan, 1908, 432-433.
• AM Ostrowski : On Some Generalizations of the Cauchy-Frullani Integral. In: Proceedings of the National Academy of Sciences . Volume 35, Number 10, October 1949, pp. 612-616, PMID 16588938 , PMC 1063092 (free full text).
• Francesco G. Tricomi : On the theorem of Frullani . American Mathematical Monthly, 58 , 1951, pages 158-164.
• AM Ostrowski : On Cauchy-Frullani Integrals , Commentarii Mathematici Helvetici 51 (1976), 57-91. doi : 10.1007 / BF02568143 (Coll. Math. Papers 349–383)
• Juan Arias-de-Reyna, On the Theorem of Frullani (PDF; 884 kB), Proc. AMS, 109: 165-175 (1990).
• Matthew Albano, Tewodros Amdeberhan, Erin Beyerstedt, and Victor H. Moll, The integrals in Gradshteyn and Ryzhik. Part 15: Frullani integrals (PDF; 106 kB), 2010, 7 pages.