CalculationOfRiemannStieltjesIntegral 2009-05-09 13:25:56 2009-05-15 13:51:28 Topic Riemann-Stieltjes integral % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} \begin{itemize} \item If $f$ is defined on\, $[a,\,b]$\, and $g$ is a constant function, then $$\int_a^bf\,dg \;=\; 0.$$ \item Let $f$ be continuous on\, $[a,\,b]$,\; $a < c < b$\, and\, $g$ the step function defined as $$g(x) = k \quad \mbox{for\;\;} x < c, \quad g(x) = k\!+\!\alpha \quad \mbox{for\;\;} x > c.$$ Then $$\int_a^bf\,dg \;=\; f(c)\cdot\alpha.$$ \item 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 $\alpha$.\, Then $g$ is sum of a continuous function $g^*$ and a step function $$h(x) = 0 \quad \mbox{for\;\;} x < c, \quad h(x) = \alpha \quad \mbox{for\;\;} x > c,$$ and one has $$ \int_a^bf\,dg \;=\; \int_a^bf\,d(g^*\!+\!h) \;=\; \int_a^bf\,dg^*+\int_a^bf\,dh \;=\; \int_a^bf\,dg^*+f(c)\cdot\alpha. $$ \item 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 \begin{align} \int_a^bf\,dg \;=\; \int_a^bf\,dg^*+\sum_if(c_i)\cdot\alpha_i. \end{align} \item 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. \end{itemize} \textbf{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 \begin{align*} g(x) \;:=\; \begin{cases} -x^2 \quad\mbox{for}\;\;\; x \leqq -2,\\ x \qquad\mbox{for}\;\; -\!2 < x \leqq 3,\\ 2x\!+\!1 \;\;\mbox{for}\;\; x > 3. \end{cases} \end{align*} 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 \begin{align*} I &\;=\; \int_{-3}^6\!f\,dg^*+f(-2)\cdot2+f(3)\cdot4 \;=\; \sum_{i=-3}^5\int_i^{i+1}\!f(x)g'(x)\,dx+0\cdot2+0\cdot4\\ &\;=\; \int_{-3}^{-2}(x\!+\!3)(-2x)\,dx+\int_{-2}^{-1}(x\!+\!2)\cdot1\,dx+\int_{-1}^0(x\!+\!1)\cdot1\,dx+ \int_0^1x\cdot1\,dx+\int_1^2(x\!-\!1)\cdot1\,dx\\ & \qquad\qquad\qquad +\int_2^3(x\!-\!2)\cdot1\,dx+\int_3^4(x\!-\!3)\cdot2\,dx+\int_4^5(x\!-\!4)\cdot2\,dx+\int_5^6(x\!-\!5)\cdot2\,dx\\ &\;=\; \frac{47}{6}. \end{align*}