Frullani type integral

  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]
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,964
    Staff Emeritus
    Science Advisor

    Did you mean to say
    [tex]G(t)= \int dt g(t) [/tex]
    in your last line?
     
  4. 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. HallsofIvy

    HallsofIvy 40,964
    Staff Emeritus
    Science Advisor

    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.
     
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?