summaryrefslogtreecommitdiff
path: root/info/mathtrip/src/misc.tex
diff options
context:
space:
mode:
Diffstat (limited to 'info/mathtrip/src/misc.tex')
-rw-r--r--info/mathtrip/src/misc.tex105
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}
+ }%
+}