ASN Inverse Sine

ASN.1 Introduction

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

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

ASN.1.1

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

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

ASN.1.2

Related functions: Inverse Cosine,Inverse Hyperbolic Cosine

ASN.2 Series and asymptotic expansions

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ASN.2.1 Asymptotic expansion at

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ASN.2.1.1 First terms

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

ASN.2.1.1.1

ASN.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).

ASN.2.2 Asymptotic expansion at $\infty$

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ASN.2.2.1 First terms

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

ASN.2.2.1.1

ASN.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).

ASN.2.3 Taylor expansion at

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ASN.2.3.1 First terms

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

ASN.2.3.1.1

ASN.2.3.2 General form

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

ASN.2.3.2.1

The coefficients $u (n)$

satisfy the recurrence

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

ASN.2.3.2.2

Initial conditions of ASN.2.3.2.2 are given by

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

ASN.2.3.2.3

ASN.2.4 Asymptotic expansion at

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ASN.2.4.1 First terms

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

ASN.2.4.1.1

ASN.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).

ASN.3 Graphs

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ASN.3.1 Real axis

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ASN.3.2 Complex plane

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