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Stieltjes integral

  1. May 3, 2006 #1
    Can you give me a simple real-life problem, where you need to use Stieltjes integral and can you show how you proceed in solving this kind of problems?
  2. jcsd
  3. May 4, 2006 #2


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    I'm not certain what you consider "real life"! I suppose it wouldn't help for me to point out that in "real life" you have to take a calculus test.

    The Stieljes integral differs from the ordinary Riemann integral in that, after we have divided the interval (a to b, say) into n intervals with endpoints xi, xi+1, instead of defining [itex]Delta xi to be simply xi+1- xi, that is, the length of the interval, we define it to be [itex]\alpha(x_{i+1})- \alpha(x_i)[/itex] where [itex]\alpha(x)[/itex] can be any increasing function. Taking the "Riemann sums" as usual then and taking the limit as the number of intervals goes to infinity results in the Stieltjes integral [itex]\int f(x)d\alpha[/itex] rather than the Riemann integral [itex]\int f(x)dx[/itex].

    Of course if [itex]\alpha(x)[/itex] happens to be differentiable then it is easy to see that
    [tex]\int f(x)d\alpha= \int f(x)\alpha'(x)dx[/tex]

    One common application is this: let [itex]\alpha(x)[/itex] be the "step" function (f(x)= 0 for 0<= x< 1, f(x)= 1 for 1<= x< 2, etc.). Then the sum
    [tex]\Sum_{n=0}^\k f(n)[/tex]
    can be written as the Stieltjes integral
    [tex]\int_0^{n+1} f(x)d\alpha[/itex]
    allowing one to combine the theory of sums with integrals.

    In particular
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