From 097f244236b682cc77c6ecf25be4150091d8daf9 Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
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}
+   }
+}
-- 
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