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ASC Inverse Secant

ASC.1 Introduction

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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 $-1$

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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 $0$

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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 $1$

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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
 
 
 
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