Frullani Formulas

Frullani's formulas relate to finding improper Riemann integrals of the form:

\int \limits _{0}^{\infty }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx

{\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {f (\ alpha x) -f (\ beta x)} {x}} \, dx}

{\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {f (\ alpha x) -f (\ beta x)} {x}} \, dx}

to which, with the help of elementary transformation, differentiation, and integration with respect to a parameter, many other improper integrals can be reduced.

Content

1 Frullani Formulas
- 1.1 The first Frullani formula
- 1.2 The second Frullani formula
- 1.3 Third Frullani Formula
2 Examples
3 notes
4 See also
5 Links

Frullani Formulas

Frullani's First Formula

If $f(x)\in C[0,+\infty )\$ ${\ displaystyle f (x) \ in C [0, + \ infty) \}$ $f(x)\in C[0,+\infty )\$ and $\ \forall A>0\ \exists \int \limits _{A}^{\infty }{\frac {f(x)}{x}}\,dx$ ${\ displaystyle \ \ forall A> 0 \ \ exists \ int \ limits _ {A} ^ {\ infty} {\ frac {f (x)} {x}} \, dx}$ $\ \forall A>0\ \exists \int \limits _{A}^{\infty }{\frac {f(x)}{x}}\,dx$ , then the following formula is valid:

\int \limits _{0}^{\infty }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx=f(0)\ln {\biggl (}{\frac {\beta }{\alpha }}{\biggr )}\ \ (\alpha >0,\beta >0)\

{\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {f (\ alpha x) -f (\ beta x)} {x}} \, dx = f (0) \ ln {\ biggl (} {\ frac {\ beta} {\ alpha}} {\ biggr)} \ \ (\ alpha> 0, \ beta> 0) \}

\int \limits _{0}^{\infty }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx=f(0)\ln {\biggl (}{\frac {\beta }{\alpha }}{\biggr )}\ \ (\alpha >0,\beta >0)\

Evidence:

\lim \limits _{A\to +0}{\Biggl (}{\int \limits _{A}^{\infty }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx}{\Biggr )}=\lim \limits _{A\to +0}{\Biggl (}{{\int \limits _{A}^{\infty }{\frac {f(\alpha x)}{x}}\,dx}-{\int \limits _{A}^{\infty }{\frac {f(\beta x)}{x}}\,dx}}{\Biggr )}=

{\ displaystyle \ lim \ limits _ {A \ to +0} {\ Biggl (} {\ int \ limits _ {A} ^ {\ infty} {\ frac {f (\ alpha x) -f (\ beta x )} {x}} \, dx} {\ Biggr)} = \ lim \ limits _ {A \ to +0} {\ Biggl (} {{\ int \ limits _ {A} ^ {\ infty} {\ frac {f (\ alpha x)} {x}} \, dx} - {\ int \ limits _ {A} ^ {\ infty} {\ frac {f (\ beta x)} {x}} \, dx }} {\ Biggr)} =}

\lim \limits _{A\to +0}{\Biggl (}{\int \limits _{A}^{\infty }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx}{\Biggr )}=\lim \limits _{A\to +0}{\Biggl (}{{\int \limits _{A}^{\infty }{\frac {f(\alpha x)}{x}}\,dx}-{\int \limits _{A}^{\infty }{\frac {f(\beta x)}{x}}\,dx}}{\Biggr )}=

=\left\{\forall A>0\ \exists \int \limits _{A}^{\infty }{\frac {f(x)}{x}}\,dx\ \Rightarrow {\int \limits _{A}^{\infty }{\frac {f(x)}{x}}\,dx=F(\infty )-F(A)}\Rightarrow {\int \limits _{A}^{\infty }{\frac {f(\alpha x)}{x}}\,dx=\int \limits _{\alpha A}^{\infty }{\frac {f(x)}{x}}\,dx}=F(\infty )-F(\alpha A)\right\}

{\ displaystyle = \ left \ {\ forall A> 0 \ \ exists \ int \ limits _ {A} ^ {\ infty} {\ frac {f (x)} {x}} \, dx \ \ Rightarrow {\ int \ limits _ {A} ^ {\ infty} {\ frac {f (x)} {x}} \, dx = F (\ infty) -F (A)} \ Rightarrow {\ int \ limits _ {A } ^ {\ infty} {\ frac {f (\ alpha x)} {x}} \, dx = \ int \ limits _ {\ alpha A} ^ {\ infty} {\ frac {f (x)} { x}} \, dx} = F (\ infty) -F (\ alpha A) \ right \}}

=\left\{\forall A>0\ \exists \int \limits _{A}^{\infty }{\frac {f(x)}{x}}\,dx\ \Rightarrow {\int \limits _{A}^{\infty }{\frac {f(x)}{x}}\,dx=F(\infty )-F(A)}\Rightarrow {\int \limits _{A}^{\infty }{\frac {f(\alpha x)}{x}}\,dx=\int \limits _{\alpha A}^{\infty }{\frac {f(x)}{x}}\,dx}=F(\infty )-F(\alpha A)\right\}

^[one]

=

{\ displaystyle =}

=

=\lim \limits _{A\to +0}{\Biggl (}F(\infty )-F(\alpha A)-F(\infty )+F(\beta A){\Biggr )}=\lim \limits _{A\to +0}{\Biggl (}F(\beta A)-F(\alpha A){\Biggr )}=\lim \limits _{A\to +0}{\Biggl (}\int \limits _{\alpha }^{\beta }{\frac {f(Ax)}{x}}\,dx{\Biggr )}=

{\ displaystyle = \ lim \ limits _ {A \ to +0} {\ Biggl (} F (\ infty) -F (\ alpha A) -F (\ infty) + F (\ beta A) {\ Biggr) } = \ lim \ limits _ {A \ to +0} {\ Biggl (} F (\ beta A) -F (\ alpha A) {\ Biggr)} = \ lim \ limits _ {A \ to +0} {\ Biggl (} \ int \ limits _ {\ alpha} ^ {\ beta} {\ frac {f (Ax)} {x}} \, dx {\ Biggr)} =}

=\lim \limits _{A\to +0}{\Biggl (}F(\infty )-F(\alpha A)-F(\infty )+F(\beta A){\Biggr )}=\lim \limits _{A\to +0}{\Biggl (}F(\beta A)-F(\alpha A){\Biggr )}=\lim \limits _{A\to +0}{\Biggl (}\int \limits _{\alpha }^{\beta }{\frac {f(Ax)}{x}}\,dx{\Biggr )}=

=\lim \limits _{A\to +0}{\Biggl (}f(A\xi )\int \limits _{\alpha }^{\beta }{\frac {1}{x}}\,dx{\Biggr )}

{\ displaystyle = \ lim \ limits _ {A \ to +0} {\ Biggl (} f (A \ xi) \ int \ limits _ {\ alpha} ^ {\ beta} {\ frac {1} {x} } \, dx {\ Biggr)}}

=\lim \limits _{A\to +0}{\Biggl (}f(A\xi )\int \limits _{\alpha }^{\beta }{\frac {1}{x}}\,dx{\Biggr )}

^[2]

=\lim \limits _{A\to +0}{\Biggl (}f(A\xi ){\biggl (}\ln(\beta )-\ln(\alpha ){\biggr )}{\Biggr )}

{\ displaystyle = \ lim \ limits _ {A \ to +0} {\ Biggl (} f (A \ xi) {\ biggl (} \ ln (\ beta) - \ ln (\ alpha) {\ biggr)} {\ Biggr)}}

=\lim \limits _{A\to +0}{\Biggl (}f(A\xi ){\biggl (}\ln(\beta )-\ln(\alpha ){\biggr )}{\Biggr )}

^[3]

=\lim \limits _{A\to +0}{\biggl (}f(A\xi ){\biggr )}\ln {\biggl (}{\frac {\beta }{\alpha }}{\biggr )}=

{\ displaystyle = \ lim \ limits _ {A \ to +0} {\ biggl (} f (A \ xi) {\ biggr)} \ ln {\ biggl (} {\ frac {\ beta} {\ alpha} } {\ biggr)} =}

=\lim \limits _{A\to +0}{\biggl (}f(A\xi ){\biggr )}\ln {\biggl (}{\frac {\beta }{\alpha }}{\biggr )}=

=\left\{\xi \in [\alpha ,\beta ]\Rightarrow \lim \limits _{A\to +0}{A\xi }=0,f(x)\in C[0,+\infty )\Rightarrow \lim \limits _{A\to +0}{f(A\xi )=f(0)}\right\}=f(0)\ln {\biggl (}{\frac {\beta }{\alpha }}{\biggr )}.

{\ displaystyle = \ left \ {\ xi \ in [\ alpha, \ beta] \ Rightarrow \ lim \ limits _ {A \ to +0} {A \ xi} = 0, f (x) \ in C [0 , + \ infty) \ Rightarrow \ lim \ limits _ {A \ to +0} {f (A \ xi) = f (0)} \ right \} = f (0) \ ln {\ biggl (} {\ frac {\ beta} {\ alpha}} {\ biggr)}.}

Frullani's Second Formula

If $f(x)\in C[0,+\infty )$ ${\ displaystyle f (x) \ in C [0, + \ infty)}$ and $\exists \lim \limits _{x\to +\infty }f(x)<+\infty \$ ${\ displaystyle \ exists \ lim \ limits _ {x \ to + \ infty} f (x) <+ \ infty \}$ then the following formula is valid:

\int \limits _{0}^{\infty }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx=(f(0)-f(+\infty ))\ln {\biggl (}{\frac {\beta }{\alpha }}{\biggr )}\ \ (\alpha >0,\beta >0)

{\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {f (\ alpha x) -f (\ beta x)} {x}} \, dx = (f (0) -f ( + \ infty)) \ ln {\ biggl (} {\ frac {\ beta} {\ alpha}} {\ biggr)} \ \ (\ alpha> 0, \ beta> 0)}

Evidence:

\int \limits _{0}^{\infty }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx=\lim \limits _{\epsilon \to 0,\Delta \to \infty }{\Biggl (}\int \limits _{\epsilon }^{A}{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx+\int \limits _{A}^{\Delta }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx{\Biggr )}

{\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {f (\ alpha x) -f (\ beta x)} {x}} \, dx = \ lim \ limits _ {\ epsilon \ to 0, \ Delta \ to \ infty} {\ Biggl (} \ int \ limits _ {\ epsilon} ^ {A} {\ frac {f (\ alpha x) -f (\ beta x)} {x} } \, dx + \ int \ limits _ {A} ^ {\ Delta} {\ frac {f (\ alpha x) -f (\ beta x)} {x}} \, dx {\ Biggr)}}

^[four]

=

{\ displaystyle =}

=\left\{\rho {\bigl (}\epsilon ,A{\bigr )}<\infty ,{\frac {f(x)}{x}}\in C[\epsilon ,A]\Rightarrow \int \limits _{\epsilon }^{A}{\frac {f(x)}{x}}\,dx=F(A)-F(\epsilon )\Rightarrow \int \limits _{\epsilon }^{A}{\frac {f(\alpha x)}{x}}\,dx=F(\alpha A)-F(\alpha \epsilon )\right\}

{\ displaystyle = \ left \ {\ rho {\ bigl (} \ epsilon, A {\ bigr)} <\ infty, {\ frac {f (x)} {x}} \ in C [\ epsilon, A] \ Rightarrow \ int \ limits _ {\ epsilon} ^ {A} {\ frac {f (x)} {x}} \, dx = F (A) -F (\ epsilon) \ Rightarrow \ int \ limits _ { \ epsilon} ^ {A} {\ frac {f (\ alpha x)} {x}} \, dx = F (\ alpha A) -F (\ alpha \ epsilon) \ right \}}

^[one]

=

{\ displaystyle =}

=\lim \limits _{\epsilon \to 0,\Delta \to +\infty }{\Biggl (}F(\alpha A)-F(\alpha \epsilon )-F(\beta A)+F(\beta \epsilon )+\int \limits _{A}^{\Delta }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx{\Biggr )}=

{\ displaystyle = \ lim \ limits _ {\ epsilon \ to 0, \ Delta \ to + \ infty} {\ Biggl (} F (\ alpha A) -F (\ alpha \ epsilon) -F (\ beta A) + F (\ beta \ epsilon) + \ int \ limits _ {A} ^ {\ Delta} {\ frac {f (\ alpha x) -f (\ beta x)} {x}} \, dx {\ Biggr )} =}

=\left\{\rho {\bigl (}A,\Delta {\bigr )}<\infty ,{\frac {f(x)}{x}}\in C[A,\Delta ]\Rightarrow \int \limits _{A}^{\Delta }{\frac {f(x)}{x}}\,dx=F(\Delta )-F(A)\Rightarrow \int \limits _{A}^{\Delta }{\frac {f(\alpha x)}{x}}\,dx=F(\alpha \Delta )-F(\alpha A)\right\}=

{\ displaystyle = \ left \ {\ rho {\ bigl (} A, \ Delta {\ bigr)} <\ infty, {\ frac {f (x)} {x}} \ in C [A, \ Delta] \ Rightarrow \ int \ limits _ {A} ^ {\ Delta} {\ frac {f (x)} {x}} \, dx = F (\ Delta) -F (A) \ Rightarrow \ int \ limits _ { A} ^ {\ Delta} {\ frac {f (\ alpha x)} {x}} \, dx = F (\ alpha \ Delta) -F (\ alpha A) \ right \} =}

=\lim \limits _{\epsilon \to +0,\Delta \to +\infty }{\Biggl (}F(\alpha A)-F(\alpha \epsilon )-F(\beta A)+F(\beta \epsilon )+F(\alpha \Delta )-F(\alpha A)-F(\beta \Delta )+F(\beta A){\Biggr )}=

{\ displaystyle = \ lim \ limits _ {\ epsilon \ to +0, \ Delta \ to + \ infty} {\ Biggl (} F (\ alpha A) -F (\ alpha \ epsilon) -F (\ beta A ) + F (\ beta \ epsilon) + F (\ alpha \ Delta) -F (\ alpha A) -F (\ beta \ Delta) + F (\ beta A) {\ Biggr)} =}

=\lim \limits _{\epsilon \to +0}{\biggl (}F(\beta \epsilon )-F(\alpha \epsilon ){\biggr )}-\lim \limits _{\Delta \to +\infty }{\biggl (}F(\beta \Delta )-F(\alpha \Delta ){\biggr )}=\lim \limits _{\epsilon \to +0}{\Biggl (}\int \limits _{\alpha }^{\beta }{\frac {f(\epsilon x)}{x}}\,dx{\Biggr )}-\lim \limits _{\Delta \to +\infty }{\Biggl (}\int \limits _{\alpha }^{\beta }{\frac {f(\Delta x)}{x}}\,dx{\Biggr )}=

{\ displaystyle = \ lim \ limits _ {\ epsilon \ to +0} {\ biggl (} F (\ beta \ epsilon) -F (\ alpha \ epsilon) {\ biggr)} - \ lim \ limits _ {\ Delta \ to + \ infty} {\ biggl (} F (\ beta \ Delta) -F (\ alpha \ Delta) {\ biggr)} = \ lim \ limits _ {\ epsilon \ to +0} {\ Biggl ( } \ int \ limits _ {\ alpha} ^ {\ beta} {\ frac {f (\ epsilon x)} {x}} \, dx {\ Biggr)} - \ lim \ limits _ {\ Delta \ to + \ infty} {\ Biggl (} \ int \ limits _ {\ alpha} ^ {\ beta} {\ frac {f (\ Delta x)} {x}} \, dx {\ Biggr)} =}

=\lim \limits _{\epsilon \to +0}{\Biggl (}f(\epsilon \eta )\int \limits _{\alpha }^{\beta }{\frac {1}{x}}\,dx{\Biggr )}-\lim \limits _{\Delta \to +\infty }{\Biggl (}f(\Delta \mu )\int \limits _{\alpha }^{\beta }{\frac {1}{x}}\,dx{\Biggr )}

{\ displaystyle = \ lim \ limits _ {\ epsilon \ to +0} {\ Biggl (} f (\ epsilon \ eta) \ int \ limits _ {\ alpha} ^ {\ beta} {\ frac {1} { x}} \, dx {\ Biggr)} - \ lim \ limits _ {\ Delta \ to + \ infty} {\ Biggl (} f (\ Delta \ mu) \ int \ limits _ {\ alpha} ^ {\ beta} {\ frac {1} {x}} \, dx {\ Biggr)}}

^[2]

={\biggl (}\lim \limits _{\epsilon \to +0}f(\epsilon \eta )-\lim \limits _{\Delta \to +\infty }f(\Delta \mu ){\biggr )}{\biggl (}\ln(\beta )-\ln(\alpha ){\biggr )}

{\ displaystyle = {\ biggl (} \ lim \ limits _ {\ epsilon \ to +0} f (\ epsilon \ eta) - \ lim \ limits _ {\ Delta \ to + \ infty} f (\ Delta \ mu ) {\ biggr)} {\ biggl (} \ ln (\ beta) - \ ln (\ alpha) {\ biggr)}}

^[3]

=

{\ displaystyle =}

=\left\{\eta ,\mu \in [\alpha ,\beta ]\Rightarrow \lim \limits _{\epsilon \to +0}\epsilon \eta =0,\lim \limits _{\Delta \to +\infty }\Delta \mu =+\infty ,f(x)\in C[0,+\infty ]\Rightarrow \lim \limits _{\epsilon \to +0}f(\epsilon \eta )=f(0),\lim \limits _{\Delta \to +\infty }f(\Delta \mu )=f(+\infty )\right\}=

{\ displaystyle = \ left \ {\ eta, \ mu \ in [\ alpha, \ beta] \ Rightarrow \ lim \ limits _ {\ epsilon \ to +0} \ epsilon \ eta = 0, \ lim \ limits _ { \ Delta \ to + \ infty} \ Delta \ mu = + \ infty, f (x) \ in C [0, + \ infty] \ Rightarrow \ lim \ limits _ {\ epsilon \ to +0} f (\ epsilon \ eta) = f (0), \ lim \ limits _ {\ Delta \ to + \ infty} f (\ Delta \ mu) = f (+ \ infty) \ right \} =}

={\biggl (}f(0)-f(+\infty ){\biggr )}\ln {\biggl (}{\frac {\beta }{\alpha }}{\biggr )}.

{\ displaystyle = {\ biggl (} f (0) -f (+ \ infty) {\ biggr)} \ ln {\ biggl (} {\ frac {\ beta} {\ alpha}} {\ biggr)}. }

Frullani's Third Formula

If $f(x)\in C(0,+\infty )\$ ${\ displaystyle f (x) \ in C (0, + \ infty) \}$ and $\ \forall A>0\ \exists \int \limits _{0}^{A}{\frac {f(x)}{x}}\,dx$ ${\ displaystyle \ \ forall A> 0 \ \ exists \ int \ limits _ {0} ^ {A} {\ frac {f (x)} {x}} \, dx}$ and $\exists \lim \limits _{x\to +\infty }f(x)<+\infty \$ ${\ displaystyle \ exists \ lim \ limits _ {x \ to + \ infty} f (x) <+ \ infty \}$ then the following formula is valid:

\int \limits _{0}^{\infty }{\frac {f(\alpha x)-f(\beta x)}{x}}\,dx=f(+\infty )\ln {\biggl (}{\frac {\alpha }{\beta }}{\biggr )}\ \ (\alpha >0,\beta >0)\

{\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {f (\ alpha x) -f (\ beta x)} {x}} \, dx = f (+ \ infty) \ ln {\ biggl (} {\ frac {\ alpha} {\ beta}} {\ biggr)} \ \ (\ alpha> 0, \ beta> 0) \}

Examples

$\int \limits _{0}^{\infty }{\frac {{\frac {\sin(\alpha x)}{\alpha x}}-{\frac {\sin(\beta x)}{\beta x}}}{x}}\,dx\,=\,\ln \left({\frac {\beta }{\alpha }}\right)$ ${\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {{\ frac {\ sin (\ alpha x)} {\ alpha x}} - {\ frac {\ sin (\ beta x) } {\ beta x}}} {x}} \, dx \, = \, \ ln \ left ({\ frac {\ beta} {\ alpha}} \ right)}$

$\int \limits _{0}^{\infty }\!{\frac {\sin \left(\alpha x+m\right)-\sin \left(\beta x+m\right)}{x}}{dx}=\sin(m)\cdot \ln \left({\frac {\beta }{\alpha }}\right)$ ${\ displaystyle \ int \ limits _ {0} ^ {\ infty} \! {\ frac {\ sin \ left (\ alpha x + m \ right) - \ sin \ left (\ beta x + m \ right)} {x}} {dx} = \ sin (m) \ cdot \ ln \ left ({\ frac {\ beta} {\ alpha}} \ right)}$

$\int \limits _{0}^{\infty }\!{\frac {\cos \left(\alpha x+m\right)-\cos \left(\beta x+m\right)}{x}}{dx}=\cos(m)\cdot \ln \left({\frac {\beta }{\alpha }}\right)$ {\ displaystyle \ int \ limits _ {0} ^ {\ infty} \! {\ frac {\ cos \ left (\ alpha x + m \ right) - \ cos \ left (\ beta x + m \ right)} {x}} {dx} = \ cos (m) \ cdot \ ln \ left ({\ frac {\ beta} {\ alpha}} \ right)}

$\int \limits _{0}^{\infty }{\frac {{\frac {m}{n+\alpha x}}-{\frac {m}{n+\beta x}}}{x}}\,dx\,=\,{{\frac {m}{n}}\,\ln \left({\frac {\beta }{\alpha }}\right)}$ ${\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {{\ frac {m} {n + \ alpha x}} - {\ frac {m} {n + \ beta x}}} {x }} \, dx \, = \, {{\ frac {m} {n}} \, \ ln \ left ({\ frac {\ beta} {\ alpha}} \ right)}}$

$\int \limits _{0}^{\infty }\!\,{\frac {{\frac {arctg\left(-\alpha \,x\right)}{\alpha \,x}}-{\frac {arctg\left(-\beta \,x\right)}{\beta \,x}}}{x}}{dx}\,={\,\ln \left({\frac {\alpha }{\beta }}\right)}$ ${\ displaystyle \ int \ limits _ {0} ^ {\ infty} \! \, {\ frac {{\ frac {arctg \ left (- \ alpha \, x \ right)} {\ alpha \, x}} - {\ frac {arctg \ left (- \ beta \, x \ right)} {\ beta \, x}}} {x}} {dx} \, = {\, \ ln \ left ({\ frac { \ alpha} {\ beta}} \ right)}}$

Notes

↑ ¹ ² Antiderivatives
↑ ¹ ² Average theorem in a definite integral
↑ ¹ ² Table of integrals
↑ Riemann integral , linearity property

Links

Weisstein, Eric W. Frullani's Integral on the Wolfram MathWorld website.
Prudnikov A.P. , Brychkov Yu.A. , Marichev O.I. Integrals and series. - M .: Nauka, 1981 .-- 800 p.

[Первообразная-1] ¹ ² Antiderivatives

[ReferenceA-2] ¹ ² Average theorem in a definite integral

[Таблица_интегралов-3] ¹ ² Table of integrals

[4] Riemann integral , linearity property

Frullani Formulas

Content

Frullani Formulas

Frullani's First Formula

Frullani's Second Formula

Frullani's Third Formula

Examples

Notes

See also

Links

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