Frullani integral

Summary

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 :

Proof for continuously differentiable functions

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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  .

Applications

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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.[1]

References

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  • G. Boros, Victor Hugo Moll, Irresistible Integrals (2004), pp. 98
  • Juan Arias-de-Reyna, On the Theorem of Frullani (PDF; 884 kB), Proc. A.M.S. 109 (1990), 165-175.
  • ProofWiki, proof of Frullani's integral.
  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. Retrieved 17 June 2020.