ASC Inverse Secant

ASC.1 Introduction

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

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

ASC.1.1

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

Related functions: Inverse Hyperbolic Secant,Inverse Cosecant

ASC.2 Series and asymptotic expansions

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ASC.2.1 Asymptotic expansion at

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ASC.2.1.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arcsec} (x)\approx (\pi\ldots) + \sqrt{x + 1} \biggl(-i\sqrt{2} - \frac{5 i}{12} (x + 1) \sqrt{2} - \frac{43 i}{160} (x + 1)^{2} \sqrt{2} \\ & \quad{}\quad{}- \frac{177 i}{896} (x + 1)^{3} \sqrt{2} - \frac{2867 i}{18432} (x + 1)^{4} \sqrt{2} - \\ & \quad{}\quad{}\frac{11531 i}{90112} (x + 1)^{5} \sqrt{2} - \frac{92479 i}{851968} (x + 1)^{6} \sqrt{2} - \\ & \quad{}\quad{}\frac{74069 i}{786432} (x + 1)^{7} \sqrt{2} - \frac{11857475 i}{142606336} (x + 1)^{8} \sqrt{2}\ldots\biggr). \end{split} \end{equation*}$

ASC.2.1.1.1

ASC.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).

ASC.2.2 Asymptotic expansion at

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ASC.2.2.1 First terms

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

ASC.2.2.1.1

ASC.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).

ASC.2.3 Asymptotic expansion at

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ASC.2.3.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arcsec} (x)\approx \sqrt{x - 1} \Biggl(\sqrt{2} - \frac{5 \sqrt{2} (x - 1)}{12} + \frac{43 \sqrt{2} (x - 1)^{2}}{160} - \\ & \quad{}\quad{}\frac{177 \sqrt{2} (x - 1)^{3}}{896} + \frac{2867 \sqrt{2} (x - 1)^{4}}{18432} - \frac{11531 \sqrt{2} (x - 1)^{5}}{90112} + \\ & \quad{}\quad{}\frac{92479 \sqrt{2} (x - 1)^{6}}{851968} - \frac{74069 \sqrt{2} (x - 1)^{7}}{786432} + \\ & \quad{}\quad{}\frac{11857475 \sqrt{2} (x - 1)^{8}}{142606336} - \frac{47442055 \sqrt{2} (x - 1)^{9}}{637534208} + \\ & \quad{}\quad{}\frac{126527543 \sqrt{2} (x - 1)^{10}}{1879048192} - \frac{1518418695 \sqrt{2} (x - 1)^{11}}{24696061952} + \\ & \quad{}\quad{}\frac{24295375159 \sqrt{2} (x - 1)^{12}}{429496729600} - \frac{97182800711 \sqrt{2} (x - 1)^{13}}{1855425871872} + \\ & \quad{}\quad{}\frac{777467420263 \sqrt{2} (x - 1)^{14}}{15942918602752} - \frac{3109879375897 \sqrt{2} (x - 1)^{15}}{68169720922112}\ldots\Biggr). \end{split} \end{equation*}$

ASC.2.3.1.1

ASC.2.3.2 General form

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

ASC.2.3.2.1

The coefficients $u (n)$

satisfy the recurrence

$\begin{equation*} \begin{split} & 2 u (n) \biggl(\frac{1}{2} + n\biggr) n + u (n - 1) \biggl(-\frac{1}{2} + n\biggr) \biggl(-\frac{1}{2} + 3 n\biggr) + u (n - 2) \biggl(-\frac{3}{2} + n\biggr) \biggl(-\frac{1}{2} + n\biggr) \\ & \quad{}\quad{}=0. \end{split} \end{equation*}$

ASC.2.3.2.2

Initial conditions of ASC.2.3.2.2 are given by

$\begin{equation*} \begin{split} u (0)& =\sqrt{2}, \\ u (1)& =\frac{-5\sqrt{2}}{12}. \end{split} \end{equation*}$

ASC.2.3.2.3