CI Cosine Integral

CI.1 Introduction

Let $x$ be a complex variable of $\mathbb{C} \setminus \{0,\infty\}$ .The function Cosine Integral (noted $\operatorname{Ci}$ ) is defined by the following third order differential equation

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

CI.1.1

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

Related function: Sine Integral

CI.2 Series and asymptotic expansions

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CI.2.1 Asymptotic expansion at

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CI.2.1.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{Ci} (x)\approx \Biggl(\frac{x^{2}}{4} - \frac{x^{4}}{96} + \frac{x^{6}}{4320} - \frac{x^{8}}{322560} - \operatorname{ln} (x) + \gamma\ldots\Biggr). \end{split} \end{equation*}$

CI.2.1.1.1

CI.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).

CI.2.2 Asymptotic expansion at $\infty$

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CI.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{Ci} (x)\approx \operatorname{e} ^{\biggl(-\frac{\operatorname{RootOf} _{\xi,1} (1 + \xi^{2})}{x}\biggr)} x y _{0} (x) + \\ & \quad{}\quad{}\operatorname{e} ^{\biggl(-\frac{\operatorname{RootOf} _{\xi,2} (1 + \xi^{2})}{x}\biggr)} x y _{1} (x), \end{split} \end{equation*}$

where

$\begin{equation*} \begin{split} y _{0} (x)& =\frac{i}{2} + \frac{i}{2} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) x + \\ & \quad{}\quad{}\biggl(\frac{i}{4} + \frac{5 i}{4} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\biggr) x^{2} + \Biggl(i \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) + \\ & \quad{}\quad{}4 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \biggl(\frac{i}{4} + \frac{5 i}{4} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\biggr)\Biggr) x^{3} + 2 \ldots \\ y _{1} (x)& =\frac{-i}{2} - \frac{i}{2} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) x - \\ & \quad{}\quad{}-\biggl(\frac{-i}{4} - \frac{5 i}{4} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\biggr) x^{2} - -\Biggl(-i\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) + \\ & \quad{}\quad{}4 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \biggl(\frac{-i}{4} - \frac{5 i}{4} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\biggr)\Biggr) x^{3} + \\ & \quad{}\quad{}2 \ldots \end{split} \end{equation*}$

CI.2.2.2 General form

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CI.2.2.2.1 Auxiliary function $y _{0} (x)$

The coefficients $u (n)$

of $y _{0} (x)$ satisfy the following recurrence

$\begin{equation*} \begin{split} & 2 u (n) n + u (n - 1) \Bigl(-2\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) - \\ & 5 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) (n - 1) - 3 (n - 1)^{2} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr) + \\ & u (n - 2) \bigl(8 - 5 n - 4 (n - 2)^{2} - (n - 2)^{3}\bigr)=0 \end{split} \end{equation*}$

whose initial conditions are given by

$\begin{equation*} \begin{split} u (1)& =\frac{i}{2} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \\ u (0)& =\frac{i}{2} \end{split} \end{equation*}$

CI.2.2.2.2 Auxiliary function $y _{1} (x)$

The coefficients $u (n)$

of $y _{1} (x)$ satisfy the following recurrence

$\begin{equation*} \begin{split} & 2 u (n) n + u (n - 1) \Bigl(-2\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) - \\ & 5 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) (n - 1) - 3 (n - 1)^{2} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr) + \\ & u (n - 2) \bigl(8 - 5 n - 4 (n - 2)^{2} - (n - 2)^{3}\bigr)=0 \end{split} \end{equation*}$

whose initial conditions are given by

$\begin{equation*} \begin{split} u (1)& =-\frac{i}{2}\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \\ u (0)& =\frac{-i}{2} \end{split} \end{equation*}$