From 097f244236b682cc77c6ecf25be4150091d8daf9 Mon Sep 17 00:00:00 2001 From: Norbert Preining Date: Sun, 2 Oct 2022 03:04:14 +0000 Subject: CTAN sync 202210020304 --- info/mathtrip/src/finite.tex | 78 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 78 insertions(+) create mode 100644 info/mathtrip/src/finite.tex (limited to 'info/mathtrip/src/finite.tex') diff --git a/info/mathtrip/src/finite.tex b/info/mathtrip/src/finite.tex new file mode 100644 index 0000000000..594dbc372b --- /dev/null +++ b/info/mathtrip/src/finite.tex @@ -0,0 +1,78 @@ +%This macro provides the text about the finite calculus +%on the first horiyontal area of the page 8. + +%The macro has one parameter +% 1( The width of the mathematical text +\newcommand\TEightFiniteCalculusA[1]{% + \deflength{\HSpace}{#1}% + \parbox[t]{\HSpace}{% + \TEightSeriesFiniteCalculusFontSize + \DisplaySpace{\TEightDisplaySpace}{\TEightDisplayShortSpace} + + \TEightTitle{Difference, shift operators:} + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulaeB}{\SmallChar}{\StyleWithoutNumber} + \Fm{\Delta f(x) = f(x+1) - f(x)i} + \Fm{\E f(x) = f(x+1)} + \end{DisplayFormulae} + + \TEightTitle{Fundamental Theorem:} + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulaeB}{\SmallChar}{\StyleWithoutNumber} + \Fm{f(x) = \Delta F(x) \Leftrightarrow \sum f(x) \delta x = F(x) + C} + \Fm{\sum_a^b f(x) \delta x = \sum_{i=a}^{b-1} f(i)} + \end{DisplayFormulae} + + \TEightTitle{Differences:} + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulaeB}{\SmallChar}{\StyleWithoutNumber} + \Fm{\Delta(c u) = c \Delta u} + \Fm{\Delta(u + v) = \Delta u + \Delta v} + \Fm{\Delta(uv) = u\Delta v + \E v \Delta u} + \Fm{\Delta(x^{\underline n}) = n x^{\underline n - 1}} + \Fm{\Delta(H_x) = x^{\underline{-1}}} + \Fm{\Delta(2^x) = 2^x} + \Fm{\Delta(c^x) = (c-1)c^x} + \Fm{\Delta\binom{x}{m} = \binom{x}{m-1}.} + \end{DisplayFormulae} + + \TEightTitle{Sums:} + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulaeB}{\SmallChar}{\StyleWithoutNumber} + \Fm{\sum c u \, \delta x = c \sum u \, \delta x} + \Fm{\sum (u + v) \, \delta x = \sum u \, \delta x + \sum v \, \delta x} + \Fm{\sum u \Delta v \, \delta x = uv - \sum\E v \Delta u \, \delta x} + \Fm{\sum x^{\underline n}\, \delta x = \frac{x^{\underline{n + 1}}}{m + 1}} + \Fm{\sum x^{\underline{-1}} \, \delta x = H_x} + \Fm{\sum c^x \, \delta x = {c^x \over c-1}} + \Fm{\sum \binom{x}{m} \, \delta x = \binom{x}{m+1}} + \end{DisplayFormulae} + + \TEightTitle{Falling Factorial Powers:} + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulaeB}{\SmallChar}{\StyleWithoutNumber} + \Fm{x^{\underline n} = x(x-1) \cdots (x-n+1), \quad n > 0} + \Fm{x^{\underline 0} = 1} + \Fm{x^{\underline n} = \frac{1}{(x+1) \cdots (x+\vert n \vert)}, \quad n < 0} + \Fm{x^{\underline{n+m}} = x^{\underline m}(x-m)^{\underline n}} + \end{DisplayFormulae} + + \TEightTitle{Rising Factorial Powers:} + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulaeB}{\SmallChar}{\StyleWithoutNumber} + \Fm{x^{\overline n} = x(x+1) \cdots (x+n-1), \quad n > 0} + \Fm{x^{\overline 0} = 1} + \Fm{x^{\overline n} = \frac{1}{(x-1) \cdots (x-\vert n \vert)}, \quad n < 0} + \Fm{x^{\overline{n+m}} = x^{\overline m}(x+m)^{\overline n}} + \end{DisplayFormulae} + + \TEightTitle{Conversion:} + \AdjustSpace{-2ex plus .5ex minus .5ex}% + \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulaeB}{\SmallChar}{\StyleWithoutNumber} + \def\FirstPart{x^{\underline n}=} + \FmPartA{\FirstPart (-1)^n(-x)^{\overline n}=(x-n+1)^{\overline n}=} + \FmPartB{\FirstPart}{1/(x+1)^{\overline{-n}}} + \def\FirstPart{x^{\overline n} =} + \FmPartA{\FirstPart (-1)^n(-x)^{\underline n}=(x+n-1)^{\underline n}=} + \FmPartB{\FirstPart}{1/(x-1)^{\underline{-n}}} + \Fm{x^n = \sum_{k=1}^n \SousEnsemble{n}{k} x^{\underline k}= \sum_{k=1}^n + \SousEnsemble{n}{k} (-1)^{n-k} x^{\overline k}} + \Fm{x^{\underline n} = \sum_{k=1}^n \Cycle{n}{k} (-1)^{n-k} x^k} + \Fm{x^{\overline n} = \sum_{k=1}^n \Cycle{n}{k} x^k} + \end{DisplayFormulae} + } +} -- cgit v1.2.3