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%This command provides the two last columns of
%the second horizontal part of the page 10
%
%The macro has one parameter:
% 1) The width available to typeset the formulae
\newcommand\TTenStieltjes[1]{%
%This command print one line of the array showing
%the differents Stieltjes integrals.
%
%The command has two parameters:
% 1) The part of equation before the equal sign
% 2) The part of the equation following it
\def\LineOfArray##1##2{%
##1&=&##2\\[\TTenExpansionSkip]%
}
\parbox[t]{#1}{%
\TTenStieljesFontSize
\DisplaySpace{\TTenDisplaySpace}{\TTenDisplayShortSpace}
\noindent
If $G$ is continuous in the interval $[a,b]$ and $F$ is nondecreasing then
\begin{displaymath}
\int_a^b G(x) \, d F(x)
\end{displaymath}
exists.
\AdjustSpace{1.5ex plus .5ex minus 1ex}
If $a \leq b \leq c$ then
\begin{displaymath}
\int_a^c G(x) \, d F(x) = \int_a^b G(x) \, d F(x) + \int_b^c G(x) \, d F(x)
\end{displaymath}
\AdjustSpace{1.5ex plus .5ex minus 1ex}
If the integrals involved exist
\begin{displaymath}
\begin{array}{lcl}
\LineOfArray{\int_a^b \big(G(x) + H(x)\big)\, d F(x)}%
{\int_a^b G(x) \, d F(x) + \int_a^b H(x) \, d F(x)}%
\LineOfArray{\int_a^b G(x)\, d \big(F(x) + H(x)\big)}%
{\int_a^b G(x) \, d F(x) + \int_a^b G(x) \, d H(x)}%
\LineOfArray{\int_a^b c \cdot G(x)\, d F(x)}%
{\int_a^b G(x)\, d \big(c \cdot F(x)\big) = c \int_a^b G(x) \, d F(x)}%
\LineOfArray{\int_a^b G(x)\, d F(x)}%
{G(b)F(b) - G(a)F(a) - \int_a^b F(x) \, d G(x)}%
\end{array}
\end{displaymath}
If the integrals involved exist,
and $F$ possesses a derivative $F'$ at every point in $[a,b]$ then
\begin{displaymath}
\int_a^b G(x) \, d F(x) = \int_a^b G(x) F'(x) \, dx
\end{displaymath}
}%
}
%The title of this part
\newcommand\TTenTitleStieltjes{Stieltjes Integration}
|