Frullani integral

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In mathematics, Frullani integrals (also called Cauchy–Frullani integrals) are a specific type of improper integral named after the Italian mathematician Giuliano Frullani (and the french mathematician Augustin Cauchy). The integrals are of the form ∫ 0 ∞ f ( a x ) − f ( b x ) x d x {\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x} where f {\displaystyle f} is a function defined for all non-negative real numbers that has a limit at ∞ {\displaystyle \infty } , which we denote by f ( ∞ ) {\displaystyle f(\infty )} . The formula below appears without proof in a letter from Frullani dated 1821 and was proved by Cauchy in 1823. The following formula for their general solution holds if f {\displaystyle f} is continuous on ( 0 , ∞ ) {\displaystyle (0,\infty )} , has finite limit at ∞ {\displaystyle \infty } , and a , b > 0 {\displaystyle a,b>0} : ∫ 0 ∞ f ( a x ) − f ( b x ) x d x = ( f ( ∞ ) − f ( 0 ) ) ln ⁡ a b . {\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x={\Big (}f(\infty )-f(0){\Big )}\ln {\frac {a}{b}}.} If f ( ∞ ) {\displaystyle f(\infty )} does not exist, but ∫ c ∞ f ( x ) x d x {\displaystyle \int _{c}^{\infty }{\frac {f(x)}{x}}dx} exists for some c > 0 {\displaystyle c>0} , then ∫ 0 ∞ f ( a x ) − f ( b x ) x d x = − f ( 0 ) ln ⁡ a b . {\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x=-f(0)\ln {\frac {a}{b}}.}