HASN Inverse Hyperbolic Sine

HASN.1 Introduction

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

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

HASN.1.1

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

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

HASN.1.2

HASN.2 Series and asymptotic expansions

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HASN.2.1 Asymptotic expansion at

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HASN.2.1.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arcsinh} (x)\approx \biggl(-\frac{i}{2}\pi\ldots\biggr) + \sqrt{x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)} \Biggl(1 + i - \\ & \quad{}\quad{}\frac{(1 + i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)}{12 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)} + \\ & \quad{}\quad{}\frac{(3 + 3 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{2}}{160 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}} - \\ & \quad{}\quad{}\frac{(5 + 5 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{3}}{896 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{3}} + \\ & \quad{}\quad{}\frac{(35 + 35 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{4}}{18432 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4}} - \\ & \quad{}\quad{}\frac{(63 + 63 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{5}}{90112 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{5}} + \\ & \quad{}\quad{}\frac{(231 + 231 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{6}}{851968 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{6}} - \\ & \quad{}\quad{}\frac{(143 + 143 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{7}}{1310720 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{7}} + \\ & \quad{}\quad{}\frac{(6435 + 6435 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{8}}{142606336 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{8}}\ldots\Biggr). \end{split} \end{equation*}$

HASN.2.1.1.1

HASN.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).

HASN.2.2 Asymptotic expansion at $\infty$

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HASN.2.2.1 First terms

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

HASN.2.2.1.1

HASN.2.2.2 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).

HASN.2.3 Taylor expansion at

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HASN.2.3.1 First terms

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$\begin{equation*} \begin{split} \operatorname{arcsinh} (x)& =x - \frac{1}{6} x^{3} + \frac{3}{40} x^{5} - \frac{5}{112} x^{7} + \frac{35}{1152} x^{9} - \frac{63}{2816} x^{11} + \frac{231}{13312} \\ & \quad{}\quad{}x^{13} - \frac{143}{10240} x^{15} + \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$

HASN.2.3.1.1

HASN.2.3.2 General form

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

HASN.2.3.2.1

The coefficients $u (n)$

satisfy the recurrence

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

HASN.2.3.2.2

Initial conditions of HASN.2.3.2.2 are given by

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

HASN.2.3.2.3

HASN.2.4 Asymptotic expansion at

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HASN.2.4.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arcsinh} (x)\approx \biggl(\frac{i}{2} \pi\ldots\biggr) + \sqrt{x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)} \Biggl(1 - i - \\ & \quad{}\quad{}\frac{(1 - i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)}{12 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)} + \\ & \quad{}\quad{}\frac{(3 - 3 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{2}}{160 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}} - \\ & \quad{}\quad{}\frac{(5 - 5 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{3}}{896 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{3}} + \\ & \quad{}\quad{}\frac{(35 - 35 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{4}}{18432 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4}} - \\ & \quad{}\quad{}\frac{(63 - 63 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{5}}{90112 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{5}} + \\ & \quad{}\quad{}\frac{(231 - 231 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{6}}{851968 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{6}} - \\ & \quad{}\quad{}\frac{(143 - 143 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{7}}{1310720 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{7}} + \\ & \quad{}\quad{}\frac{(6435 - 6435 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{8}}{142606336 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{8}}\ldots\Biggr). \end{split} \end{equation*}$

HASN.2.4.1.1

HASN.2.4.2 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).

HASN.3 Graphs

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HASN.3.1 Real axis

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HASN.3.2 Complex plane

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