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'calculation of Riemann--Stieltjes integral'
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| Title of object: |
calculation of Riemann--Stieltjes integral |
| Canonical Name: |
CalculationOfRiemannStieltjesIntegral |
| Type: |
Topic |
| Created on: |
2009-05-09 13:25:56 |
| Modified on: |
2009-05-09 13:25:56 |
| Classification: |
msc:26A42 |
Revision comment (for changes between this and next version):
Preamble:
% 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}
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%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
\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
$$
\int_a^bf\,dg \;=\; \int_a^bf\,d(g^*\!+\!h) \;=\; \int_a^bf\,dg^*+\int_a^bf\,dh \;=\; \int_a^bf\,dg^*+f(c)\cdot\alpha.
$$
\end{itemize}
[Not ready . . .] |
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