HCS Hyperbolic Cosine

HCS.1 Introduction

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

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

HCS.1.1

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

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

HCS.1.2

Related function: Hyperbolic Sine

HCS.2 Series and asymptotic expansions

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HCS.2.1 Asymptotic expansion at $\infty$

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HCS.2.1.1 Exact form

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$\begin{equation*} \begin{split} \operatorname{cosh} (x)& =\frac{\operatorname{e} ^{x}}{2} - \frac{\operatorname{e} ^{(-x)}}{2}. \end{split} \end{equation*}$

HCS.2.2 Taylor expansion at

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