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Diffstat (limited to 'info/mathtrip/src/stieltjes.tex')
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diff --git a/info/mathtrip/src/stieltjes.tex b/info/mathtrip/src/stieltjes.tex new file mode 100644 index 0000000000..10ecef09f1 --- /dev/null +++ b/info/mathtrip/src/stieltjes.tex @@ -0,0 +1,54 @@ +%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} |