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HSN Hyperbolic Sine

HSN.1 Introduction

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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 $0$

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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
 
 
 
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