In mathematics, Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. 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 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
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 TonelliâÂÂs theorem to interchange the two integrals:
Note that the integral in the second line above has been taken over the interval , not .
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 ,
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.