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HACS Inverse Hyperbolic Cosine

HACS.1 Introduction

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


\begin{equation*} 
\begin{split} 
x \frac{\partial y (x)}{\partial x} + \bigl(x^{2} - 1\bigr) \frac{\partial^{2} y (x)}{\partial x^{2}}& =0. 
\end{split} 
\end{equation*}
HACS.1.1

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


\begin{equation*} 
\begin{split} 
\operatorname{arccosh} (0)& =\frac{i}{2} \pi, \\ 
\frac{\partial \operatorname{arccosh} (x)}{\partial x} (0)& =-i. 
\end{split} 
\end{equation*} 
 HACS.1.2

Related functions: Inverse Cosine,Inverse Sine

HACS.2 Series and asymptotic expansions

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HACS.2.1 Asymptotic expansion at $-1$

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HACS.2.1.1 First terms

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\begin{equation*} 
\begin{split} 
& \operatorname{arccosh} (x)\approx (i \pi\ldots) + \sqrt{x + 1} \biggl(-i\sqrt{2} - \frac{i}{12} (x + 1) \sqrt{2} -  \\ 
& \quad{}\quad{}\frac{3 i}{160} (x + 1)^{2} \sqrt{2} - \frac{5 i}{896} (x + 1)^{3} \sqrt{2} - \frac{35 i}{18432} (x + 1)^{4} \sqrt{2} -  \\ 
& \quad{}\quad{}\frac{63 i}{90112} (x + 1)^{5} \sqrt{2} - \frac{231 i}{851968} (x + 1)^{6} \sqrt{2} - \frac{143 i}{1310720} (x + 1)^{7} \sqrt{2}  \\ 
& \quad{}\quad{}- \frac{6435 i}{142606336} (x + 1)^{8} \sqrt{2}\ldots\biggr). 
\end{split} 
\end{equation*} 
 HACS.2.1.1.1

HACS.2.1.2 General form

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The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).

HACS.2.2 Asymptotic expansion at $1$

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HACS.2.2.2 General form

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\begin{equation*} 
\begin{split} 
& \operatorname{arccosh} (x)\approx \sqrt{x - 1} \sum_{n = 0}^{\infty} u (n) (x - 1)^{n}. 
\end{split} 
\end{equation*}
HACS.2.2.2.1
The coefficients $u (n)$ satisfy the recurrence

\begin{equation*} 
\begin{split} 
2 u (n) \biggl(n + \frac{1}{2}\biggr) n + u (n - 1) \biggl(-\frac{1}{2} + n\biggr)^{2}& =0. 
\end{split} 
\end{equation*}
HACS.2.2.2.2
Initial conditions of HACS.2.2.2.2 are given by

\begin{equation*} 
\begin{split} 
u (0)& =\sqrt{2}. 
\end{split} 
\end{equation*}
HACS.2.2.2.3
The recurrence HACS.2.2.2.2 has the closed form solution

\begin{equation*} 
\begin{split} 
u (n)& =\frac{2^{\bigl(n + \frac{1}{2}\bigr)} \Gamma \Bigl(n + \frac{1}{2}\Bigr) (-1)^{n}}{4^{n} \sqrt{\pi} \Gamma (n + 1) (2 n + 1)}. 
\end{split} 
\end{equation*}
HACS.2.2.2.4

HACS.2.3 Asymptotic expansion at $\infty$

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HACS.2.3.1 First terms

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\begin{equation*} 
\begin{split} 
& \operatorname{arccosh} (x)\approx \Biggl(\operatorname{ln} (2) + \frac{1}{4 x^{2}} + \frac{3}{32 x^{4}} + \frac{5}{96 x^{6}} + \frac{35}{1024 x^{8}} + \operatorname{ln} \biggl(\frac{1}{x}\biggr)\ldots\Biggr). 
\end{split} 
\end{equation*} 
 HACS.2.3.1.1

HACS.2.3.2 General form

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The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).

HACS.2.4 Taylor expansion at $0$

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HACS.2.4.2 General form

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\begin{equation*} 
\begin{split} 
\operatorname{arccosh} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}. 
\end{split} 
\end{equation*} 
 HACS.2.4.2.1
The coefficients $u (n)$ satisfy the recurrence

\begin{equation*} 
\begin{split} 
n^{2} u (n) - -\bigl(-n^{2} - 3 n - 2\bigr) u (n + 2)& =0. 
\end{split} 
\end{equation*}
HACS.2.4.2.2
Initial conditions of HACS.2.4.2.2 are given by

\begin{equation*} 
\begin{split} 
u (0)& =\frac{i}{2} \pi, \\ 
u (1)& =-i. 
\end{split} 
\end{equation*}
HACS.2.4.2.3
 
 
 
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