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%The following command typeset the series of the second
%column of the page 9
%
%This command has one parameter:
% 1) The width of the mathematical text
\newcommand\TNineSeriesThree[1]{%
%This command typeset a fact about a serie and
%the math used to describe the property.
%
%The command has two parameters:
% 1) The title of the fact
% 2) The corresponding math.
%This macro uses the global (to the macro) variables '\Hspace', '\CurrentLineWidth'
%which should contain the space used to typeset the math.
\def\SerieProperty##1##2{%
\def\temp{\TNineTitle{##1} \ensuremath{##2}}%
\savebox\TmpBoxA{\temp}%
\settowidth\CurrentLineWidth{\usebox\TmpBoxA}%
\ifdimcomp{\CurrentLineWidth}{<}{\HSpace}%
{%Enough space, everything is written on one line
\temp
}{%The title is on one line, the math typeset as 'display math'
\TNineTitle{##1}%
\begin{displaymath}%
##2%
\end{displaymath}%
}%
\par
\AdjustSpace{\TNineInterTitle}%
}%
%This command typeset a line showing the expansion of ordinary power serie
%
%The command has 2 parameters
% 1) The formula to be expanded
% 2) The expanded version
\def\LineOfOrdPowSerie##1##2{##1&=&##2\\[\TNineExpansionSkip]}%
\parbox[t]{#1}{%
\TNineSeriesFontSize
\DisplaySpace{\TNineDisplaySpace}{\TNineDisplayShortSpace}%
\setlength{\HSpace}{#1}%
\SerieProperty{Ordinary power series:}
{A(x) = \sum_{i=0}^\infty a_i x^i}
\SerieProperty{Exponential power series:}
{A(x) = \sum_{i=0}^\infty a_i \frac{x^i}{i!}}
\SerieProperty{Dirichlet power series:}
{A(x) = \sum_{i=1}^\infty \frac{a_i}{i^x}}
\SerieProperty{Binomial theorem:}
{(x + y)^n = \sum_{k=0}^n\binom{n}{k} x^{n-k} y^k}
\SerieProperty{Difference of like powers:}
{x^n - y^n = (x-y) \sum_{k=0}^{n-1} x^{n-1-k} y^k}
\SerieProperty{For ordinary power series:}
{\begin{array}{lcl}
\LineOfOrdPowSerie{\alpha A(x) + \beta B(x)}%
{\sum_{i=0}^\infty(\alpha a_i + \beta b_i) x^i}
\LineOfOrdPowSerie{x^k A(x)}%
{\sum_{i=k}^\infty a_{i-k} x^i}
\LineOfOrdPowSerie{\frac{A(x) - \sum_{i=0}^{k-1} a_i x^i}{x^k}}%
{\sum_{i=0}^\infty a_{i+k} x^i}
\LineOfOrdPowSerie{A(cx)}%
{\sum_{i=0}^\infty c^i a_i x^i}
\LineOfOrdPowSerie{A'(x)}%
{\sum_{i=0}^\infty (i+1) a_{i+1} x^i}
\LineOfOrdPowSerie{x A'(x)}%
{\sum_{i=1}^\infty i a_i x^i}
\LineOfOrdPowSerie{\int A(x) \, dx}%
{\sum_{i=1}^\infty \frac{a_{i-1}}{i} x^i}
\LineOfOrdPowSerie{\frac{A(x) + A(-x)}{2}}%
{\sum_{i=0}^\infty a_{2i} x^{2i}}
\LineOfOrdPowSerie{\frac{A(x) - A(-x)}{2}}%
{\sum_{i=0}^\infty a_{2i+1} x^{2i+1}}
\end{array}
}
\SerieProperty{Summation:}
{\text{If }b_i = \sum_{j=0}^i a_i\text{ then }
B(x) = \frac{1}{1 -x} A(x)}
\SerieProperty{Convolution:}
{A(x)B(x) = \sum_{i=0}^\infty\left(\sum_{j=0}^i a_j b_{i-j}\right)x^i}
}
}
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