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Diffstat (limited to 'info/mathtrip/src/misc.tex')
-rw-r--r-- | info/mathtrip/src/misc.tex | 105 |
1 files changed, 105 insertions, 0 deletions
diff --git a/info/mathtrip/src/misc.tex b/info/mathtrip/src/misc.tex new file mode 100644 index 0000000000..53de2a6e8a --- /dev/null +++ b/info/mathtrip/src/misc.tex @@ -0,0 +1,105 @@ +%This command provides the text about the 'General' part +%of the page 3 +% +%The command has one parameter: +% 1) The width of the text +\newcommand\TThreeGeneral[1]{% + \parbox[t]{#1}{% + \DisplaySpace{\TThreeDisplaySpace}{\TThreeDisplayShortSpace} + + %Since the columns is narrow, ragged right looks nicer + %Don't use \RaggedRight here since the result is not good + \raggedright + + \TThreeTitle{Bernoulli Numbers + (\MathRemark[\relax]{B_i = 0}, \MathRemark[\relax]{\text{odd} i \neq 1}):}% + \begin{DisplayFormulae}{1}{0pt}{5ex plus 1ex minus 1ex}{\SmallChar}{\StyleWithoutNumber} + \def\FmSep{\unskip\text{,}} + \Fm{B_0 = 1} + \Fm{B_1 = -{\frac{1}{2}}} + \Fm{B_2 = \frac{1}{6}} + \Fm{B_4 = -{\frac{1}{30}}} + \Fm{B_6 = \frac{1}{42}} + \Fm{B_8 = -{\frac{1}{30}}} + \def\FmSep{\relax} + \Fm{B_{10} = \frac{5}{66}} + \end{DisplayFormulae} + + \TThreeTitle{Change of base, quadratic formula:}% + \begin{DisplayFormulae}{1}{0pt}{6ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber} + \Fm{\log_b x = \frac{\log_a{x}}{\log_a{b}}}, + \Fm{\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}} + \end{DisplayFormulae} + + \TThreeTitle{Euler's number $e$:} + \AdjustSpace{1.5ex plus .5ex minus .5ex} + \begin{DisplayFormulae}{1}{0pt}{4ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber} + \def\FmSep{\unskip\text{,}} + \Fm{e = 1 + \tfrac{1}{2} + \tfrac{1}{6} + \tfrac{1}{24} + \cdots = \sum_{n=0}^\infty\frac{1}{n!}} + \Fm{\lim_{n \to \infty} \Bigl(1 + \frac{x}{n}\Bigr)^n = e^x} + \Fm{\left(1 + \tfrac{1}{n} \right)^n < e < \left(1 + \tfrac{1}{n} \right)^{n+1}} + \def\FmSep{\relax} + \def\FirstPart{\left(1 + \tfrac{1}{n} \right)^n = \mbox{}} + \FmPartA{\FirstPart e - \frac{e}{2 n} + \frac{11 e}{24 n^2} -} + \FmPartB{\FirstPart}{O\left(\frac{1}{n^3}\right)} + \end{DisplayFormulae} + + \TThreeTitle{Harmonic numbers:} + \AdjustSpace{1.5ex plus .5ex minus .5ex} + \begin{DisplayFormulae}{1}{0pt}{5ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber} + \def\FmSep{\unskip\text{,}} + \Fm{1} + \Fm{\frac{3}{2}} + \Fm{\frac{11}{6}} + \Fm{\frac{25}{12}} + \Fm{\frac{137}{60}} + \Fm{\frac{49}{20}} + \Fm{\frac{363}{140}} + \Fm{\frac{761}{280}} + \Fm{\frac{7129}{2520}} + \Fm{\ldots} + \Fm{\ln n < H_n < \ln n + 1} + \def\FmSep{\relax} + \Fm{H_n = \ln n + \gamma + O\bigg(\frac{1}{n} \bigg)} + \end{DisplayFormulae} + + \TThreeTitle{Factorial, Stirling's approximation:} + \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\SmallChar}{\StyleWithoutNumber} + \def\FmSep{\unskip\text{,}} + \Fm{1} + \Fm{2} + \Fm{6} + \Fm{24} + \Fm{120} + \Fm{720} + \Fm{5040} + \Fm{40320} + \Fm{362880} + \def\FmSep{\relax} + \Fm{\ldots} + \Fm{n! = \sqrt{2\pi n} \bigg(\frac{n}{e}\bigg)^n\bigg(1 + + \Theta\bigg(\frac{1}{n}\bigg)\bigg)} + \end{DisplayFormulae} + + \TThreeTitle{Ackermann's function and inverse:} + \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\SmallChar}{\StyleWithoutNumber} + {%This equation is just too wide and not easy to split, font size + %reduction seems to be the easiest solution + %\fontsize{9.3pt}{10pt}\selectfont + \fontsize{8.8pt}{9.6pt}\selectfont + \Fm{a(i,j) = + \left\{% + \begin{array}{lr} + 2^j &\MathRemark[\relax]{i = 1} \\ + a(i-1,2) &\MathRemark[\relax]{j=1} \\ + a(i-1,a(i,j-1)) &\MathRemark[\relax]{i,j \geq 2} + \end{array} + \right.% + } + } + %Force the next formula on a new line + \NewParFormulae + \Fm{\alpha(i) = \min\{j \mid a(j,j) \geq i \}} + \end{DisplayFormulae} + }% +} |