CS Cosine

CS.1 Introduction

Let $x$ be a complex variable of $\mathbb{C} \setminus \{\infty\}$ .The function Cosine (noted $\operatorname{cos}$ ) is defined by the following second order differential equation

$\begin{equation*} \begin{split} y (x) + \frac{\partial^{2} y (x)}{\partial x^{2}}& =0. \end{split} \end{equation*}$

CS.1.1

The initial conditions of CS.1.1 are given at $0$ by

$\begin{equation*} \begin{split} \operatorname{cos} (0)& =1, \\ \frac{\partial \operatorname{cos} (x)}{\partial x} (0)& =0. \end{split} \end{equation*}$

CS.1.2

Related function: Sine

CS.2 Series and asymptotic expansions

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CS.2.1 Asymptotic expansion at $\infty$

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CS.2.1.1 Exact form

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$\begin{equation*} \begin{split} \operatorname{cos} (x)& =\frac{\operatorname{e} ^{(-\operatorname{RootOf} _{\xi,2} (1 + \xi^{2}) x)}}{2} + \frac{\operatorname{e} ^{(-\operatorname{RootOf} _{\xi,1} (1 + \xi^{2}) x)}}{2}. \end{split} \end{equation*}$

CS.2.2 Taylor expansion at

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CS.2.2.1 First terms

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$\begin{equation*} \begin{split} \operatorname{cos} (x)& =1 - \frac{1}{2} x^{2} + \frac{1}{24} x^{4} - \frac{1}{720} x^{6} + \frac{1}{40320} x^{8} - \frac{1}{3628800} x^{10} + \\ & \quad{}\quad{}\frac{1}{479001600} x^{12} - \frac{1}{87178291200} x^{14} + \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$