HSN Hyperbolic Sine

HSN.1 Introduction

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

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

HSN.1.1

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

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

HSN.1.2

Related function: Hyperbolic Cosine

HSN.2 Series and asymptotic expansions

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HSN.2.1 Taylor expansion at

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

HSN.2.1.1.1

HSN.2.1.2 General form

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$\begin{equation*} \begin{split} \operatorname{sinh} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}. \end{split} \end{equation*}$

HSN.2.1.2.1

The coefficients $u (n)$

satisfy the recurrence

$\begin{equation*} \begin{split} -u (n) + \bigl(n^{2} + 3 n + 2\bigr) u (n + 2)& =0. \end{split} \end{equation*}$

HSN.2.1.2.2

Initial conditions of HSN.2.1.2.2 are given by

$\begin{equation*} \begin{split} u (0)& =0, \\ u (1)& =1. \end{split} \end{equation*}$

HSN.2.1.2.3

The recurrence HSN.2.1.2.2 has the closed form solution

$\begin{equation*} \begin{split} u (2 n + 1)& =\frac{1}{\Gamma (2 n + 2)}, \\ u (2 n)& =0. \end{split} \end{equation*}$

HSN.2.1.2.4

HSN.2.2 Asymptotic expansion at $\infty$

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HSN.2.2.1 Exact form

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

HSN.3 Graphs

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HSN.3.1 Real axis

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HSN.3.2 Complex plane

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