SN Sine

SN.1 Introduction

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

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

SN.1.1

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

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

SN.1.2

Related function: Cosine

SN.2 Series and asymptotic expansions

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SN.2.1 Asymptotic expansion at $\infty$

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SN.2.1.1 Exact form

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

SN.2.2 Taylor expansion at

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SN.2.2.1 First terms

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$\begin{equation*} \begin{split} \operatorname{sin} (x)& =x - \frac{1}{6} x^{3} + \frac{1}{120} x^{5} - \frac{1}{5040} x^{7} + \frac{1}{362880} x^{9} - \frac{1}{39916800} x^{11} + \\ & \quad{}\quad{}\frac{1}{6227020800} x^{13} - \frac{1}{1307674368000} x^{15} + \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$