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BA1 - Analyse I/BA1 - Analyse I.tex

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+\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
+
+\input{../Base.tex}
+
+\title{Formulaire d'Analyse I}
+
+\begin{document}
+
+\begin{tabu}to \textwidth{ |X|X| }
+\hline
+
+\textbf{Polynômes de Taylor} \newline
+$ \begin{aligned}
+                          \e^x & = \sum\limits_{k = 0}^\infty \frac{x^k}{k!}, &&x \in \symbb{R} \\
+       \sinh \left( x \right) & = \sum\limits_{k = 0}^\infty \frac{x^{2k+1}}{\left( 2k+1 \right)!}, &&x \in \symbb{R} \\
+       \cosh \left( x \right) & = \sum\limits_{k = 0}^\infty \frac{x^{2k}}{\left( 2k \right)!}, &&x \in \symbb{R} \\
+        \sin \left( x \right) & = \sum\limits_{k = 0}^\infty \left( -1 \right)^k \cdot \frac{x^{2k+1}}{\left( 2k+1 \right)!}, &&x \in \symbb{R} \\
+        \cos \left( x \right) & = \sum\limits_{k = 0}^\infty \left( -1 \right)^k \cdot \frac{x^{2k}}{\left( 2k \right)!}, &&x \in \symbb{R} \\
+       \ln \left( 1+x \right) & = \sum\limits_{k = 0}^\infty \left( -1 \right)^{k+1} \cdot \frac{x^k}{k}, &&x \in \left] -1, 1 \right[ \\
+                \frac{1}{1+x} & = \sum\limits_{k = 0}^\infty \left( -1 \right)^{k} \cdot x^k, &&x \in \left] -1, 1 \right[ \\
+     \arctan \left( x \right) & = \sum\limits_{k = 0}^\infty \left( -1 \right)^k \cdot \frac{x^{2k+1}}{2k+1}, &&x \in \left] -1, 1 \right[ \\
+\end{aligned} $
+&
+\textbf{Intégrales} \newline
+$ \begin{aligned}
+  &\int \frac{f' \left( x \right)}{f \left( x \right)} \cdot \dif x && = \ln \abs{f \left( x \right)} + C \\
+  &\int \frac{f' \left( x \right)}{1+f^2 \left( x \right)} \cdot \dif x && = \arctan \left[ f \left( x \right) \right] + C \\
+  &\int \left[ f \left( x \right) \right]^\alpha \cdot f' \left( x \right) \cdot \dif x && = \frac{\left[ f \left( x \right) \right]^{\alpha+1}}{\alpha + 1} + C, &\forall \alpha \neq -1 \\
+  &\int \e^{f \left( x \right)} \cdot f' \left( x \right) \cdot \dif x && = \e^{f \left( x \right)} + C \\
+  &\int \frac{f' \left( x \right)}{\sqrt{1-f^2 \left( x \right)}} \cdot \dif x && = \arcsin \left[ f \left( x \right) \right] + C \\
+\end{aligned} $
+\\
+\end{tabu}
+
+\nointerlineskip
+
+\begin{tabu}to \textwidth{ |X|X|X| }
+\hline
+
+\textbf{Racine carrée complexe} \newline
+$ \begin{aligned}
+  w = u + v \cdot \im, z = a + b \cdot \im, z^2 = w \\
+  \begin{cases}
+    a^2 - b^2 & = u \\
+    2 \cdot a \cdot b & = 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} $
+&
+\\\hline
+
+\textbf{Exponentielle} \newline
+$ \begin{aligned}
+   \cos \left( \theta \right) & = \frac{\e^{\im \cdot \theta} + \e^{-\im \cdot \theta}}{2} \\
+   \sin \left( \theta \right) & = \frac{\e^{\im \cdot \theta} - \e^{-\im \cdot \theta}}{2 \cdot \im} \\
+  \cosh \left( \theta \right) & = \frac{\e^{\theta} + \e^{-\theta}}{2} \\
+  \sinh \left( \theta \right) & = \frac{\e^{\theta} - \e^{-\theta}}{2} \\
+\end{aligned} $
+&
+\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 \left( x \right) & = +\infty \\
+   \lim_{x \to \infty} f \left( x \right) \cdot h \left( x \right) & = L \\
+   \lim_{x \to \infty} \left[ 1 + h \left( x \right) \right]^{f \left( x \right)} & = \e^L \\
+\end{aligned} $
+&
+\textbf{Trigonométrie} \newline
+$ \begin{aligned}
+   \cosh^2 \left( x \right) - \sinh^2 \left( x \right) = 1 \\
+   \cos^2 \left( x \right) + \sin^2 \left( x \right) = 1 \\
+   \sin \left( x+y \right) = \sin \left( x \right) \cdot \cos \left( y \right) + \cos \left( x \right) \cdot \sin \left( y \right) \\
+   \cos \left( x+y \right) = \cos \left( x \right) \cdot \cos \left( y \right) + \sin \left( x \right) \cdot \sin \left( y \right) \\
+   \sin \left( x \right) + \sin \left( y \right) = 2 \cdot \sin \left( \frac{x+y}{2} \right) \cdot \cos \left( \frac{x-y}{2} \right) \\
+   \sin \left( x \right) - \sin \left( y \right) = 2 \cdot \sin \left( \frac{x-y}{2} \right) \cdot \cos \left( \frac{x+y}{2} \right) \\
+   \cos \left( x \right) + \cos \left( y \right) = 2 \cdot \cos \left( \frac{x+y}{2} \right) \cdot \cos \left( \frac{x-y}{2} \right) \\
+   \cos \left( x \right) - \cos \left( y \right) = -2 \cdot \sin \left( \frac{x+y}{2} \right) \cdot \sin \left( \frac{x-y}{2} \right) \\
+\end{aligned} $
+\\\hline
+\end{tabu}
+
+\nointerlineskip
+
+\begin{tabu}to \textwidth{ |X| }
+
+\textbf{Angles particuliers} \newline
+$ \begin{aligned}
+   \cos \left( 0 \right) = 1 \qquad &\cos \left( \frac{\pi}{6} \right) = \frac{1}{2}         \qquad &\cos \left( \frac{\pi}{4} \right) = \frac{1}{2} \cdot \sqrt{2} \qquad &\cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \cdot \sqrt{3} \qquad &\cos \left( \frac{\pi}{2} \right) = 0 \\
+   \sin \left( 0 \right) = 0 \qquad &\sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \cdot \sqrt{3} \qquad &\sin \left( \frac{\pi}{4} \right) = \frac{1}{2} \cdot \sqrt{2} \qquad &\sin \left( \frac{\pi}{3} \right) = \frac{1}{2}         \qquad &\sin \left( \frac{\pi}{2} \right) = 1 \\
+\end{aligned} $
+\\\hline
+
+\textbf{Convergence} \newline
+$ \begin{aligned}
+  &\int_M^\infty x^a \cdot \e^{-bx} \cdot \dif x   &\text{ converge pour tout } a \in \symbb{R} \text{ et tout } b > 0 \\
+  &\int_a^\infty \frac{1}{x^p} \cdot \dif x &\text{ converge si et seulement si } p > 1 \quad \left( a > 0 \right) \\
+  &\int_0^b \frac{1}{x^p} \cdot \dif x      &\text{ converge si et seulement si } p < 1 \\
+\end{aligned} $
+\\\hline
+
+\end{tabu}
+\end{document}