# calculation of Riemann–Stieltjes integral

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

 $\int_{a}^{b}f\,dg\;=\;0.$
• Let $f$ be continuous   on  $[a,\,b]$,  $a  and  $g$ the step function  defined as

 $g(x)=k\quad\mbox{for\;\;}xc.$

Then

 $\int_{a}^{b}f\,dg\;=\;f(c)\cdot\alpha.$
• Let $f$ be continuous on  $[a,\,b]$,  $a  and the function $g$ be otherwise continuous but have in  $x=c$  a step of magnitude $\alpha$.  Then $g$ is sum of a continuous function $g^{*}$ and a step function

 $h(x)=0\quad\mbox{for\;\;}xc,$

and one has

 $\int_{a}^{b}f\,dg\;=\;\int_{a}^{b}f\,d(g^{*}\!+\!h)\;=\;\int_{a}^{b}f\,dg^{*}+% \int_{a}^{b}f\,dh\;=\;\int_{a}^{b}f\,dg^{*}+f(c)\cdot\alpha.$
• 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 $\alpha_{i}$ in respectively the same points $c_{i}$ on the interval$[a,\,b]$  as the function $g$.  If $f$ is Riemann–Stieltjes integrable on  $[a,\,b]$,  then

 $\displaystyle\int_{a}^{b}f\,dg\;=\;\int_{a}^{b}f\,dg^{*}+\sum_{i}f(c_{i})\cdot% \alpha_{i}.$ (1)
• Suppose that  $g=g^{*}\!+\!h$ (as above) has a finite amount of steps $\alpha_{i}$ in the points $c_{i}$ 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

 $I\;:=\;\int_{-3}^{6}(x\!-\!\lfloor{x}\rfloor)\,dg(x)$

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

 $\displaystyle g(x)\;:=\;\begin{cases}-x^{2}\quad\mbox{for}\;\;\;x\leqq-2,\\ x\qquad\mbox{for}\;\;-\!23.\end{cases}$

Now, $f$ is from the left discontinuous  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

 $g\;=\;g^{*}\!+\!h$

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

 $\displaystyle I$ $\displaystyle\;=\;\int_{-3}^{6}\!f\,dg^{*}+f(-2)\cdot 2+f(3)\cdot 4\;=\;\sum_{% i=-3}^{5}\int_{i}^{i+1}\!f(x)g^{\prime}(x)\,dx+0\cdot 2+0\cdot 4$ $\displaystyle\;=\;\int_{-3}^{-2}(x\!+\!3)(-2x)\,dx+\int_{-2}^{-1}(x\!+\!2)% \cdot 1\,dx+\int_{-1}^{0}(x\!+\!1)\cdot 1\,dx+\int_{0}^{1}x\cdot 1\,dx+\int_{1% }^{2}(x\!-\!1)\cdot 1\,dx$ $\displaystyle\qquad\qquad\qquad+\int_{2}^{3}(x\!-\!2)\cdot 1\,dx+\int_{3}^{4}(% x\!-\!3)\cdot 2\,dx+\int_{4}^{5}(x\!-\!4)\cdot 2\,dx+\int_{5}^{6}(x\!-\!5)% \cdot 2\,dx$ $\displaystyle\;=\;\frac{47}{6}.$
Title calculation of Riemann–Stieltjes integral CalculationOfRiemannStieltjesIntegral 2013-03-22 18:55:09 2013-03-22 18:55:09 pahio (2872) pahio (2872) 8 pahio (2872) Topic msc 26A42