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Revision difference : calculation of Riemann--Stieltjes integral |
| Version 4 |
Version 3 |
| \begin{itemize} |
\begin{itemize} |
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| \item If $f$ is defined on\, $[a,\,b]$\, and $g$ is a constant function, then |
\item If $f$ is defined on\, $[a,\,b]$\, and $g$ is a constant function, then |
| $$\int_a^bf\,dg \;=\; 0.$$ |
$$\int_a^bf\,dg \;=\; 0.$$ |
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| \item Let $f$ be continuous on\, $[a,\,b]$,\; $a < c < b$\, and\, $g$ the step function defined as |
\item Let $f$ be continuous on\, $[a,\,b]$,\; $a < c < b$\, and\, $g$ the step function defined as |
| $$g(x) = k \quad \mbox{for\;\;} x < c, \quad g(x) = k\!+\!\alpha \quad \mbox{for\;\;} x > c.$$ |
$$g(x) = k \quad \mbox{for\;\;} x < c, \quad g(x) = k\!+\!\alpha \quad \mbox{for\;\;} x > c.$$ |
| Then |
Then |
| $$\int_a^bf\,dg \;=\; f(c)\cdot\alpha.$$ |
$$\int_a^bf\,dg \;=\; f(c)\cdot\alpha.$$ |
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| \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 |
\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 |
| $$h(x) = 0 \quad \mbox{for\;\;} x < c, \quad h(x) = \alpha \quad \mbox{for\;\;} x > c,$$ |
$$h(x) = 0 \quad \mbox{for\;\;} x < c, \quad h(x) = \alpha \quad \mbox{for\;\;} x > c,$$ |
| and one has |
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. |
\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. |
| $$ |
$$ |
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| \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 |
\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} |
\begin{align} |
| \int_a^bf\,dg \;=\; \int_a^bf\,dg^*+\sum_if(c_i)\cdot\alpha_i. |
\int_a^bf\,dg \;=\; \int_a^bf\,dg^*+\sum_if(c_i)\cdot\alpha_i. |
| \end{align} |
\end{align} |
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| \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. |
\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 |
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$f$ 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. |
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| \end{itemize} |
\end{itemize} |
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| \textbf{Example.}\, Find the value of the Riemann--Stieltjes integral |
[Not ready . . .] |
| $$I \;:=\; \int_{-3}^6(x\!-\!\lfloor{x}\rfloor)\,dg(x)$$ |
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| where the integrand $f$ is the mantissa function and the integrator $g$ defined by |
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| \begin{align*} |
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| g(x) \;:=\; |
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| \begin{cases} |
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| -x^2 \quad\mbox{for}\;\;\; x \leqq -2,\\ |
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| x \qquad\mbox{for}\;\; -\!2 < x \leqq 3,\\ |
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| 2x\!+\!1 \;\;\mbox{for}\;\; x > 3. |
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| \end{cases} |
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| \end{align*} |
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| 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 |
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| $$g \;=\; g^*\!+\!h$$ |
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| where $g^*$ is continuous and the step function $h$ has the step of 2 at $-2$ and the step of 4 at 3.\; We get |
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| \begin{align*} |
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| 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\\ |
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| &\;=\; \int_{-3}^{-2}(x\!+\!3)(-2x)\,dx+\int_{-2}^{-1}(x\!+\!2)\cdot1\,dx+\int_{-1}^0(x\!+\!1)\cdot1\,dx+ |
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| \int_0^1x\cdot1\,dx+\int_1^2(x\!-\!1)\cdot1\,dx\\ |
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| & \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\\ |
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| &\;=\; \frac{47}{6}. |
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| \end{align*} |
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