calculation of Riemann–Stieltjes integral

  • If f is defined on  [a,b]  and g is a constant function, then

    abf𝑑g= 0.
  • Let f be continuousMathworldPlanetmathPlanetmath on  [a,b],  a<c<b  and  g the step functionPlanetmathPlanetmath defined as

    g(x)=kfor x<c,g(x)=k+αfor x>c.


  • Let f be continuous on  [a,b],  a<c<b  and the function g be otherwise continuous but have in  x=c  a step of magnitude α.  Then g is sum of a continuous function g* and a step function

    h(x)=0for x<c,h(x)=αfor x>c,

    and one has

  • Suppose that g can be expressed in the form  g=g*+h  where g* is continuous and h a step function having an at most denumerable amount of steps αi in respectively the same points ci on the interval[a,b]  as the function g.  If f is Riemann–Stieltjes integrable on  [a,b],  then

    abf𝑑g=abf𝑑g*+if(ci)αi. (1)
  • Suppose that  g=g*+h (as above) has a finite amount of steps αi in the points ci of the interval  [a,b]  but f does not have same-sided discontinuities as g in any of those points.  Then f is Riemann–Stieltjes integrable on the interval and the equation (1) is true.

Example.  Find the value of the Riemann–Stieltjes integral


where the integrand f is the mantissa function and the integrator g defined by

g(x):={-x2forx-2,x  for-2<x3,2x+1forx>3.

Now, f is from the left discontinuousMathworldPlanetmath at every integer, but g is bounded and only discontinuous from the right at -2 and 3.  By the above last item, f is Riemann–Stieltjes integrable with respect to g on  [-3, 6].  We can set


where g* is continuous and the step function h has the step of 2 at -2 and the step of 4 at 3.  Using (1) we get

I =-36f𝑑g*+f(-2)2+f(3)4=i=-35ii+1f(x)g(x)𝑑x+02+04
Title calculation of Riemann–Stieltjes integral
Canonical name CalculationOfRiemannStieltjesIntegral
Date of creation 2013-03-22 18:55:09
Last modified on 2013-03-22 18:55:09
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Topic
Classification msc 26A42