diff --git a/tests/image/test.tex b/tests/image/test.tex index 13cb0f2b..dda1104e 100644 --- a/tests/image/test.tex +++ b/tests/image/test.tex @@ -1,5 +1,5 @@ \documentclass{article} -\usepackage{graphicx, amsmath, amssymb, multirow} +\usepackage{graphics, amsmath, amssymb, multirow} \usepackage{geometry} \geometry{ a4paper, @@ -55,7 +55,7 @@ \section{Some beautiful mathematical equations} Pythagora's theorem: -$$a^2+b^2=c^2+d^2$$ +$$a^2+b^2=c^2$$ Logarithms: @@ -172,7 +172,7 @@ \section{Some beautiful mathematical equations} ax^2+bx+c=0 \implies x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \] -Two more ways to calculate pi: +Four more ways to calculate pi: \begin{equation*} \pi=\sum_{k=0}^\infty\left[\frac{1}{16^k}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6} \right) \right] @@ -182,8 +182,25 @@ \section{Some beautiful mathematical equations} \frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\ldots \end{equation*} +\begin{equation*} +\pi = 3+\textstyle \cfrac{1}{7+\textstyle \cfrac{1}{15+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{292+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{1+\ddots}}}}}}} +\end{equation*} + +Chudnovsky Formula: + +\[ +\frac{1}{\pi} = \frac{\sqrt{10005}}{4270934400} \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!\,k!^3 (-640320)^{3k}} +\] + Cauchy's integral formula: $$f(a)=\frac{1}{2\pi i}\int_{C} \frac{f(z)}{z-a} dz $$ + +Stirling's factorial approximation: +$$n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left( \frac{1}{n} \right) \right)$$ + \end{document} + + +