ACT Inverse Cotangent

ACT.1 Introduction

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

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

ACT.1.1

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

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

ACT.1.2

ACT.2 Series and asymptotic expansions

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ACT.2.1 Asymptotic expansion at

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ACT.2.1.1 First terms

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

ACT.2.1.1.1

ACT.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).

ACT.2.2 Asymptotic expansion at

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ACT.2.2.1 First terms

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

ACT.2.2.1.1

ACT.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).

ACT.2.3 Taylor expansion at

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ACT.2.3.1 First terms

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$\begin{equation*} \begin{split} \operatorname{arccot} (x)& =\frac{\pi}{2} - x + \frac{1}{3} x^{3} - \frac{1}{5} x^{5} + \frac{1}{7} x^{7} - \frac{1}{9} x^{9} + \frac{1}{11} x^{11} - \frac{1}{13} x^{13} + \frac{1}{15} x^{15} \\ & \quad{}\quad{}+ \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$