AHCS Inverse Hyperbolic Cosecant

AHCS.1 Introduction

Let $x$ be a complex variable of $\mathbb{C} \setminus \{0\}$ .The function Inverse Hyperbolic Cosecant (noted $\operatorname{arccsch}$ ) is defined by the following second order differential equation

$\begin{equation*} \begin{split} \bigl(2 x^{2} + 1\bigr) \frac{\partial y (x)}{\partial x} + \bigl(x^{3} + x\bigr) \frac{\partial^{2} y (x)}{\partial x^{2}}& =0. \end{split} \end{equation*}$

AHCS.1.1

The initial conditions of AHCS.1.1 at $0$ are not simple to state, since $0$ is a (regular) singular point.

AHCS.2 Series and asymptotic expansions

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AHCS.2.1 Asymptotic expansion at

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AHCS.2.1.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arccsch} (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{}\biggl(\frac{5}{12} - \frac{5 i}{12}\biggr) \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr) + (1 - i) \\ & \quad{}\quad{}\left(\frac{3}{40} + \frac{11 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}}{32}\right) \Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)^{2} + \\ & \quad{}\quad{}(1 - i) \Biggl(\frac{25 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)}{336} + \\ & \quad{}\quad{}\frac{17 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \bigl(12 + 55 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{2688}\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)^{3} + (1 - i) \Biggl(\frac{7}{384} + \\ & \quad{}\quad{}\frac{385 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}}{4608} + \\ & \quad{}\quad{}\frac{23 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} \bigl(404 + 935 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{55296}\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)^{4} + (1 - i) \Biggl(\frac{15 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)}{704} + \\ & \quad{}\quad{}\frac{51 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \bigl(12 + 55 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{28160} + 29 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \\ & \quad{}\quad{} \Bigl(1008 + 13912 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}21505 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4}\Bigr)\bigg/1351680\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)^{5} + (1 - i) \Biggl(\frac{77}{13312} + \\ & \quad{}\quad{}\frac{4235 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}}{159744} + \\ & \quad{}\quad{}\frac{253 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} \bigl(404 + 935 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{1916928} + \\ & \quad{}\quad{}7 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} \Bigl(87408 + 538088 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}623645 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4}\Bigr)\bigg/7667712\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)^{6} + (1 - i) \Biggl(\frac{13 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)}{1792} + \\ & \quad{}\quad{}\frac{221 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \bigl(12 + 55 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{358400} + 377 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \\ & \quad{}\quad{} \Bigl(1008 + 13912 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}21505 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4}\Bigr)\bigg/51609600 + 41 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \Bigl( \\ & \quad{}\quad{}44352 + 1223984 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}4712836 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4} + 4365515 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{6}\Bigr)\bigg/ \\ & \quad{}\quad{}247726080\Biggr) \Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)^{7} + (1 - i) \Biggl(\frac{1155}{557056} + \\ & \quad{}\quad{}\frac{21175 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}}{2228224} + \\ & \quad{}\quad{}\frac{1265 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} \bigl(404 + 935 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{26738688} \\ & \quad{}\quad{}+ 35 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} \Bigl(87408 + \\ & \quad{}\quad{}538088 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} + 623645 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4}\Bigr)\bigg/ \\ & \quad{}\quad{}106954752 + 47 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} \Bigl(36363456 + \\ & \quad{}\quad{}418800176 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}1160708620 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4} + \\ & \quad{}\quad{}894930575 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{6}\Bigr)\bigg/44920995840\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)^{8}\ldots\Biggr). \end{split} \end{equation*}$

AHCS.2.1.1.1

AHCS.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).

AHCS.2.2 Asymptotic expansion at

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AHCS.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arccsch} (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{}\biggl(\frac{5}{12} + \frac{5 i}{12}\biggr) \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr) + (1 + i) \\ & \quad{}\quad{}\left(\frac{3}{40} + \frac{11 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}}{32}\right) \Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)^{2} + \\ & \quad{}\quad{}(1 + i) \Biggl(\frac{25 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)}{336} + \\ & \quad{}\quad{}\frac{17 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \bigl(12 + 55 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{2688}\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)^{3} + (1 + i) \Biggl(\frac{7}{384} + \\ & \quad{}\quad{}\frac{385 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}}{4608} + \\ & \quad{}\quad{}\frac{23 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} \bigl(404 + 935 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{55296}\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)^{4} + (1 + i) \Biggl(\frac{15 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)}{704} + \\ & \quad{}\quad{}\frac{51 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \bigl(12 + 55 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{28160} + 29 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \\ & \quad{}\quad{} \Bigl(1008 + 13912 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}21505 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4}\Bigr)\bigg/1351680\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)^{5} + (1 + i) \Biggl(\frac{77}{13312} + \\ & \quad{}\quad{}\frac{4235 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}}{159744} + \\ & \quad{}\quad{}\frac{253 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} \bigl(404 + 935 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{1916928} + \\ & \quad{}\quad{}7 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} \Bigl(87408 + 538088 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}623645 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4}\Bigr)\bigg/7667712\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)^{6} + (1 + i) \Biggl(\frac{13 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)}{1792} + \\ & \quad{}\quad{}\frac{221 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \bigl(12 + 55 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{358400} + 377 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \\ & \quad{}\quad{} \Bigl(1008 + 13912 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}21505 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4}\Bigr)\bigg/51609600 + 41 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \Bigl( \\ & \quad{}\quad{}44352 + 1223984 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}4712836 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4} + 4365515 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{6}\Bigr)\bigg/ \\ & \quad{}\quad{}247726080\Biggr) \Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)^{7} + (1 + i) \Biggl(\frac{1155}{557056} + \\ & \quad{}\quad{}\frac{21175 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}}{2228224} + \\ & \quad{}\quad{}\frac{1265 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} \bigl(404 + 935 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\bigr)}{26738688} \\ & \quad{}\quad{}+ 35 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} \Bigl(87408 + \\ & \quad{}\quad{}538088 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} + 623645 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4}\Bigr)\bigg/ \\ & \quad{}\quad{}106954752 + 47 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} \Bigl(36363456 + \\ & \quad{}\quad{}418800176 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2} + \\ & \quad{}\quad{}1160708620 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4} + \\ & \quad{}\quad{}894930575 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{6}\Bigr)\bigg/44920995840\Biggr) \\ & \quad{}\quad{}\Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)^{8}\ldots\Biggr). \end{split} \end{equation*}$