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Frullani type integral

  1. May 11, 2008 #1
    under what condition the does the equality hold ?

    [tex] \int_{0}^{\infty} dt \frac{ f(at)-f(bt)}{g(t)}= (G(b)-G(a))(f(0)-f(\infty)) [/tex]

    and [tex] \int dt g(t) [/tex]
     
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  3. May 11, 2008 #2

    HallsofIvy

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    Did you mean to say
    [tex]G(t)= \int dt g(t) [/tex]
    in your last line?
     
  4. May 11, 2008 #3
    yes i meant the integral of g(t) ,i have been looking for generalizations of this integral but without any luck , under that condition can define g(t) and f(at) f(bt) so the integral above converge ?? thank
     
  5. May 12, 2008 #4

    HallsofIvy

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    The generalized mean value theorem says that if f and g are differentiable on (a, b) and continuous on [a, b], then there exist c in (a, b) such that
    [tex]\frac{f(b)- f(a)}{g(b)- g(a)}= \frac{f'(c)}{g'(c)}[/tex]

    That looks like an integral version to me.
     
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