%Produce the formulae in the first columns of the second horizontal
%part of page 10
%
%The macrohas one parameter:
% 1) The size available to typeset the math
\newcommand\TTenSerieFour[1]{%
\parbox[t]{#1}{%
\TTenSeriesFontSize
%This command typeset a line equation about the series on page 10
%when the serie is in one column
%
%The command has 2 parameters
% 1) The first formula to be expanded
% 2) its expanded version
\def\LineOfArray##1##2{%
##1&=&##2\\[\TTenExpansionSkip]%
}
%To have some longer vertical space between some lines
%of the array (to better separate some equation).
\deflength\VSpace{1.2\TTenExpansionSkip}
$\displaystyle
\begin{array}{l@{\hspace{.1em}}c@{\hspace{.2em}}l}
\LineOfArray{\zeta(x)}
{\prod_p \frac{1}{1 - p^{-x}}}
\LineOfArray{\zeta^2(x)}
{\sum_{i=1}^\infty \frac{d(i)}{x^i}}
%The previous line is too long
& & \text{where }d(n) = \sum_{d \vert n} 1 \\[\VSpace]
\LineOfArray{\zeta(x)\zeta(x-1)}
{\sum_{i=1}^\infty \frac{S(i)}{x^i}}
%The previous line is too long
& & \text{where }S(n) = \sum_{d \vert n} d \\[\VSpace]
\LineOfArray{\zeta(2n)}
{\frac{2^{2n-1} \vert B_{2n} \vert}{(2n)!} \pi^{2n}, \quad n \in \Natural}
\LineOfArray{\frac{x}{\sin x}}
{\sum_{i=0}^\infty (-1)^{i-1}\frac{(4^i - 2) B_{2i} x^{2i}}{(2i)!}}
\LineOfArray{\left(\frac{1 - \sqrt{1-4x}}{2x}\right)^n}
{\sum_{i=0}^\infty \frac{n(2i + n - 1)!}{i! (n + i)!} x^i}
\LineOfArray{e^x \sin x}%
{\sum_{i=1}^\infty \frac{2^{i/2} \sin \tfrac{i \pi}{4}}{i!} x^i}
\LineOfArray{\frac{\sqrt{1 - \sqrt{1 -x}}}{x}}%
{\sum_{i=0}^\infty \frac{(4i)!}{16^i \sqrt{2} (2i)! (2i+1)!} x^i}
\LineOfArray{\left(\frac{\arcsin x}{x}\right)^2}%
{\sum_{i=0}^\infty \frac{ 4^i i!^2}{(i + 1)(2i +1)!} x^{2i}}
\end{array}$
}
}
%The title of this part
%\newcommand\TTenSeriesBTitle{Series (continued)}
\newcommand\TTenSeriesBTitle{\relax}
