ACS Inverse Cosine

ACS.1 Introduction

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

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

ACS.1.1

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

$\begin{equation*} \begin{split} \operatorname{arccos} (0)& =\frac{\pi}{2}, \\ \frac{\partial \operatorname{arccos} (x)}{\partial x} (0)& =-1. \end{split} \end{equation*}$

ACS.1.2

Related functions: Inverse Sine,Inverse Hyperbolic Cosine

ACS.2 Series and asymptotic expansions

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ACS.2.1 Asymptotic expansion at

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ACS.2.1.1 First terms

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

ACS.2.1.1.1

ACS.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).

ACS.2.2 Asymptotic expansion at

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ACS.2.2.1 First terms

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

ACS.2.2.1.1

ACS.2.2.2 General form

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$\begin{equation*} \begin{split} & \operatorname{arccos} (x)\approx \sqrt{x - 1} \sum_{n = 0}^{\infty} u (n) (x - 1)^{n}. \end{split} \end{equation*}$

ACS.2.2.2.1

The coefficients $u (n)$

satisfy the recurrence

$\begin{equation*} \begin{split} 2 u (n) \biggl(n + \frac{1}{2}\biggr) n + u (n - 1) \biggl(-\frac{1}{2} + n\biggr)^{2}& =0. \end{split} \end{equation*}$

ACS.2.2.2.2

Initial conditions of ACS.2.2.2.2 are given by

$\begin{equation*} \begin{split} u (0)& =i \sqrt{2}. \end{split} \end{equation*}$

ACS.2.2.2.3

The recurrence ACS.2.2.2.2 has the closed form solution

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

ACS.2.2.2.4

ACS.2.3 Asymptotic expansion at $\infty$

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ACS.2.3.1 First terms

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

ACS.2.3.1.1

ACS.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).

ACS.2.4 Taylor expansion at

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ACS.2.4.1 First terms

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$\begin{equation*} \begin{split} \operatorname{arccos} (x)& =\frac{\pi}{2} - x - \frac{1}{6} x^{3} - \frac{3}{40} x^{5} - \frac{5}{112} x^{7} - \frac{35}{1152} x^{9} - \frac{63}{2816} x^{11} - \frac{231}{13312} \\ & \quad{}\quad{}x^{13} - \frac{143}{10240} x^{15} + \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$