Formulaires/analyse-I.tex

\documentclass[a4paper,10pt]{article}
%\documentclass[a4paper,10pt]{scrartcl}

\input{common.tex}

\begin{document}

\begin{tabularx}{\textwidth}{ |X|X| }
\hline

\textbf{Polynômes de Taylor} \newline
$\begin{aligned}
            e^x &= \sum\limits_{k=0}^\infty \frac{x^k}{k!}, &x \in \mathbb{R} \\
       \sinh(x) &= \sum\limits_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}, &x \in \mathbb{R} \\
       \cosh(x) &= \sum\limits_{k=0}^\infty \frac{x^{2k}}{(2k)!}, &x \in \mathbb{R} \\
        \sin(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!}, &x \in \mathbb{R} \\
        \cos(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!}, &x \in \mathbb{R} \\
       \ln(1+x) &= \sum\limits_{k=0}^\infty (-1)^{k+1} \frac{x^k}{k}, &x \in {]-1,1[} \\
  \frac{1}{1+x} &= \sum\limits_{k=0}^\infty (-1)^{k} x^k, &x \in {]-1,1[} \\
     \arctan(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{2k+1}, &x \in {]-1,1[} \\
\end{aligned}$ \newline
&
\textbf{Intégrales} \newline
$\begin{aligned}
  &\int \frac{f'(x)}{f(x)}\mathrm{d}x &&= \ln \left|f(x)\right| + C \\
  &\int \frac{f'(x)}{1+f^2(x)}\mathrm{d}x &&= \arctan \left[f(x)\right] + C \\
  &\int \left[f(x)\right]^\alpha f'(x) \mathrm{d}x &&= \frac{\left[f(x)\right]^{\alpha+1}}{\alpha + 1} + C, &\forall \alpha \neq -1 \\
  &\int e^{f(x)} f'(x) \mathrm{d}x &&= e^{f(x)} + C \\
  &\int \frac{f'(x)}{\sqrt{1-f^2(x)}}\mathrm{d}x &&= \arcsin \left[f(x)\right] + C \\
\end{aligned}$ \newline
\\
\end{tabularx}
\offinterlineskip

\begin{tabularx}{\textwidth}{ |X|X|X| }
\hline

\textbf{Racine carrée complexe} \newline
$\begin{aligned}
  w = u + vi, z = a + bi, z^2 = w \\
  \begin{cases}
    a^2 - b^2 &= u \\
    2ab &= v \\
    a^2 + b^2 &= \sqrt{u^2 + v^2} \\
  \end{cases}
\end{aligned}$
&
\textbf{Somme géométrique} \newline
$\begin{aligned}
       \sum\limits_{k=0}^n q^k &= \frac{1-q^{n+1}}{1-q} \\
  \sum\limits_{k=0}^\infty q^k &= \frac{1}{1-q} \\
\end{aligned}$ \newline
&
\\ \hline

\textbf{Exponentielle} \newline
$\begin{aligned}
   \cos(\theta) &= \frac{e^{i\theta} + e^{-i\theta}}{2} \\
   \sin(\theta) &= \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
  \cosh(\theta) &= \frac{e^{\theta} + e^{-\theta}}{2} \\
  \sinh(\theta) &= \frac{e^{\theta} - e^{-\theta}}{2} \\
\end{aligned}$ \newline
&
\textbf{Exponentielle} \newline
$\begin{aligned}
   \lim_{n \to \infty} \left(1 + \frac{L}{n}\right)^n &= e^L \\
   \text{De manière\ générale :} \\
   \lim_{x \to \infty} f(x) &= +\infty \\
   \lim_{x \to \infty} f(x)h(x) &= L \\
   \lim_{x \to \infty} \left[1 + h(x)\right]^{f(x)} &= e^L \\
\end{aligned}$ \newline
&
\textbf{Trigonométrie} \newline
$\begin{aligned}
   \cosh^2(x) - \sinh^2(x) = 1 \\
   \cos^2(x) + \sin^2(x) = 1 \\
   \sin(x+y) = \sin x \cos y + \cos x \sin y \\
   \cos(x+y) = \cos x \cos y + \sin x \sin y \\
   \sin x + \sin y = 2 \sin(\frac{x+y}{2})\cos(\frac{x-y}{2}) \\
   \sin x - \sin y = 2 \sin(\frac{x-y}{2})\cos(\frac{x+y}{2}) \\
   \cos x + \cos y = 2 \cos(\frac{x+y}{2})\cos(\frac{x-y}{2}) \\
   \cos x - \cos y = -2 \sin(\frac{x+y}{2})\sin(\frac{x-y}{2}) \\
\end{aligned}$ \newline
\\ \hline

\multicolumn{2}{|X|}{
\textbf{Angles particuliers} \newline
$\begin{aligned}
   \cos(0) = 1 \quad &\cos(\frac{\pi}{6}) = \frac{1}{2}         \quad &\cos(\frac{\pi}{4}) = \frac{1}{2}\sqrt{2} \quad &\cos(\frac{\pi}{3}) = \frac{1}{2}\sqrt{3} \quad &\cos(\frac{\pi}{2}) = 0 \\
   \sin(0) = 0 \quad &\sin(\frac{\pi}{6}) = \frac{1}{2}\sqrt{3} \quad &\sin(\frac{\pi}{4}) = \frac{1}{2}\sqrt{2} \quad &\sin(\frac{\pi}{3}) = \frac{1}{2}         \quad &\sin(\frac{\pi}{2}) = 1 \\
\end{aligned}$ \newline
}
&
\\ \hline

\multicolumn{2}{|X|}{
\textbf{Convergence} \newline
$\begin{aligned}
  &\int_M^\infty x^a e^{-bx} \mathrm{d}x   &\text{ converge pour tout } a \in \mathbb{R} \text{ et tout } b > 0 \\
  &\int_a^\infty \frac{1}{x^p} \mathrm{d}x &\text{ converge si et seulement si } p > 1 \quad (a > 0) \\
  &\int_0^b \frac{1}{x^p} \mathrm{d}x      &\text{ converge si et seulement si } p < 1 \\
\end{aligned}$ \newline
}
&
\\ \hline
\end{tabularx}
\end{document}