ATN Inverse Tangent

ATN.1 Introduction

Let $x$ be a complex variable of $\mathbb{C} \setminus \{i,-i\}$ .The function Inverse Tangent (noted $\operatorname{arctan}$ ) is defined by the following second order differential equation

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

ATN.1.1

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

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

ATN.1.2

ATN.2 Series and asymptotic expansions

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ATN.2.1 Asymptotic expansion at

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ATN.2.1.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arctan} (x)\approx \Biggl(\frac{i}{2} \operatorname{ln} (2) + \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr) \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)}{4} \\ & \quad{}\quad{}- \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{2}}{16} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{3}}{48 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)} - \\ & \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{4}}{128 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{5}}{320 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{3}} - \\ & \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{6}}{768 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4}} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{7}}{1792 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{5}} - \\ & \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{8}}{4096 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{6}} + \frac{\operatorname{ln} \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)}{2}\ldots\Biggr). \end{split} \end{equation*}$

ATN.2.1.1.1

ATN.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).

ATN.2.2 Asymptotic expansion at

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ATN.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{arctan} (x)\approx \Biggl(-\frac{i}{2}\operatorname{ln} (2) + \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr) \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)}{4} \\ & \quad{}\quad{}- \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{2}}{16} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{3}}{48 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)} - \\ & \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{4}}{128 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{5}}{320 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{3}} - \\ & \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{6}}{768 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4}} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{7}}{1792 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{5}} - \\ & \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{8}}{4096 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{6}} + \frac{\operatorname{ln} \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)}{2}\ldots\Biggr). \end{split} \end{equation*}$

ATN.2.2.1.1

ATN.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).

ATN.2.3 Taylor expansion at

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ATN.2.3.1 First terms

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$\begin{equation*} \begin{split} \operatorname{arctan} (x)& =x - \frac{1}{3} x^{3} + \frac{1}{5} x^{5} - \frac{1}{7} x^{7} + \frac{1}{9} x^{9} - \frac{1}{11} x^{11} + \frac{1}{13} x^{13} - \frac{1}{15} x^{15} \\ & \quad{}\quad{}+ \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$