%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}
   }
}
