HACS Inverse Hyperbolic Cosine

HACS.1 Introduction

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

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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

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HACS.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arccosh} (x)\approx \sqrt{x - 1} \Biggl(\sqrt{2} - \frac{\sqrt{2} (x - 1)}{12} + \frac{3 \sqrt{2} (x - 1)^{2}}{160} - \\ & \quad{}\quad{}\frac{5 \sqrt{2} (x - 1)^{3}}{896} + \frac{35 \sqrt{2} (x - 1)^{4}}{18432} - \frac{63 \sqrt{2} (x - 1)^{5}}{90112} + \\ & \quad{}\quad{}\frac{231 \sqrt{2} (x - 1)^{6}}{851968} - \frac{143 \sqrt{2} (x - 1)^{7}}{1310720} + \frac{6435 \sqrt{2} (x - 1)^{8}}{142606336} - \\ & \quad{}\quad{}\frac{12155 \sqrt{2} (x - 1)^{9}}{637534208} + \frac{46189 \sqrt{2} (x - 1)^{10}}{5637144576} - \frac{88179 \sqrt{2} (x - 1)^{11}}{24696061952} \\ & \quad{}\quad{}+ \frac{676039 \sqrt{2} (x - 1)^{12}}{429496729600} - \frac{1300075 \sqrt{2} (x - 1)^{13}}{1855425871872} + \\ & \quad{}\quad{}\frac{5014575 \sqrt{2} (x - 1)^{14}}{15942918602752} - \frac{9694845 \sqrt{2} (x - 1)^{15}}{68169720922112}\ldots\Biggr). \end{split} \end{equation*}$

HACS.2.2.1.1

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

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HACS.2.4.1 First terms

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$\begin{equation*} \begin{split} \operatorname{arccosh} (x)& =\frac{i}{2} \pi - i x - \frac{i}{6} x^{3} - \frac{3 i}{40} x^{5} - \frac{5 i}{112} x^{7} - \frac{35 i}{1152} x^{9} - \frac{63 i}{2816} x^{11} - \\ & \quad{}\quad{}\frac{231 i}{13312} x^{13} - \frac{143 i}{10240} x^{15} + \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$