summaryrefslogtreecommitdiff
path: root/info/mathtrip/src/stieltjes.tex
diff options
context:
space:
mode:
Diffstat (limited to 'info/mathtrip/src/stieltjes.tex')
-rw-r--r--info/mathtrip/src/stieltjes.tex54
1 files changed, 54 insertions, 0 deletions
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}