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
where is a function defined for all non-negative real numbers that has a limit at , which we denote by .
The formula below appears without proof in a letter from Frullani dated 1821 and was proved by Cauchy in 1823. (Œuvres complètes, series 2, volume 1, pp. 275–357: p. 339)
The following formula for their general solution holds if is continuous on , has finite limit at , and :
If does not exist, but exists for some , then
Proof for continuously differentiable functions
A simple proof of the formula (under stronger assumptions than those stated above, namely ) can be arrived at by using the Fundamental theorem of calculus to express the integrand as an integral of :
and then use Tonelli’s theorem to interchange the two integrals:
Note that the integral in the second line above has been taken over the interval , not .
Ramanujan's generalization Ramanujan, using his master theorem, gave the following generalization.
Let be functions continuous on .Let and be given as above, and assume that and are continuous functions on . Also assume that and . Then, if ,
Applications The formula can be used to derive an integral representation for the natural logarithm by letting and :
The formula can also be generalized in several different ways.
References
G. Boros, Victor Hugo Moll, Irresistible Integrals (2004), pp. 98 ProofWiki, proof of Frullani's integral.