Frullani integral

In mathematics, Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. The integrals are of the form

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

where ${\displaystyle f}$ is a function defined for all non-negative real numbers that has a limit at ${\displaystyle \infty }$, which we denote by ${\displaystyle f(\infty )}$.

The following formula for their general solution holds if ${\displaystyle f}$ is continuous on ${\displaystyle (0,\infty )}$, has finite limit at ${\displaystyle \infty }$, and ${\displaystyle a,b>0}$:

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

If ${\displaystyle f(\infty )}$ does not exist, but ${\displaystyle \int _{c}^{\infty }{\frac {f(x)}{x}}dx}$ exists for some ${\displaystyle c>0}$, then $${\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x=-f(0)\ln {\frac {a}{b}}.}$$

Proof for continuously differentiable functions

A simple proof of the formula (under stronger assumptions than those stated above, namely ${\displaystyle f\in {\mathcal {C}}^{1}(0,\infty )}$) can be arrived at by using the Fundamental theorem of calculus to express the integrand as an integral of ${\displaystyle f'(xt)={\frac {\partial }{\partial t}}\left({\frac {f(xt)}{x}}\right)}$:

${\displaystyle {\begin{aligned}{\frac {f(ax)-f(bx)}{x}}&=\left[{\frac {f(xt)}{x}}\right]_{t=b}^{t=a}\,\\&=\int _{b}^{a}f'(xt)\,dt\\\end{aligned}}}$

and then use Tonelli’s theorem to interchange the two integrals:

${\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,dx&=\int _{0}^{\infty }\int _{b}^{a}f'(xt)\,dt\,dx\\&=\int _{b}^{a}\int _{0}^{\infty }f'(xt)\,dx\,dt\\&=\int _{b}^{a}\left[{\frac {f(xt)}{t}}\right]_{x=0}^{x\to \infty }\,dt\\&=\int _{b}^{a}{\frac {f(\infty )-f(0)}{t}}\,dt\\&={\Big (}f(\infty )-f(0){\Big )}{\Big (}\ln(a)-\ln(b){\Big )}\\&={\Big (}f(\infty )-f(0){\Big )}\ln {\Big (}{\frac {a}{b}}{\Big )}\\\end{aligned}}}$

Note that the integral in the second line above has been taken over the interval ${\displaystyle [b,a]}$, not ${\displaystyle [a,b]}$.

Ramanujan's generalization

Ramanujan, using his master theorem, gave the following generalization.^[1][2]

Let ${\displaystyle f,g}$ be functions continuous on ${\displaystyle [0,\infty ]}$.$${\displaystyle f(x)-f(\infty )=\sum _{k=0}^{\infty }{\frac {u(k)(-x)^{k}}{k!}}\quad {\text{ and }}\quad g(x)-g(\infty )=\sum _{k=0}^{\infty }{\frac {v(k)(-x)^{k}}{k!}}}$$Let ${\textstyle u(x)}$ and ${\textstyle v(x)}$ be given as above, and assume that ${\textstyle f}$ and ${\textstyle g}$ are continuous functions on ${\textstyle [0,\infty )}$. Also assume that ${\textstyle f(0)=g(0)}$ and ${\textstyle f(\infty )=g(\infty )}$. Then, if ${\textstyle a,b>0}$,

$${\displaystyle \lim _{n\rightarrow 0}\int _{0}^{\infty }x^{n-1}\{f(ax)-g(bx)\}dx=\{f(0)-f(\infty )\}\left\{\log \left({\frac {b}{a}}\right)+{\frac {d}{ds}}\left(\log \left({\frac {v(s)}{u(s)}}\right)\right)_{s=0}\right\}}$$

Applications

The formula can be used to derive an integral representation for the natural logarithm ${\displaystyle \ln(x)}$ by letting ${\displaystyle f(x)=e^{-x}}$ and ${\displaystyle a=1}$:

${\displaystyle {\int _{0}^{\infty }{\frac {e^{-x}-e^{-bx}}{x}}\,{\rm {d}}x={\Big (}\lim _{n\to \infty }{\frac {1}{e^{n}}}-e^{0}{\Big )}\ln {\Big (}{\frac {1}{b}}}{\Big )}=\ln(b)}$

The formula can also be generalized in several different ways.^[3]