CHI Hyperbolic Cosine Integral

CHI.1 Introduction

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

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

CHI.1.1

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

Related function: Hyperbolic Sine Integral

CHI.2 Series and asymptotic expansions

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CHI.2.1 Asymptotic expansion at

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CHI.2.1.1 First terms

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

CHI.2.1.1.1

CHI.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).

CHI.2.2 Asymptotic expansion at $\infty$

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CHI.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{Chi} (x)\approx \operatorname{e} ^{\frac{1}{x}} x y _{0} (x) + \operatorname{e} ^{\Bigl(-\frac{1}{x}\Bigr)} x y _{1} (x), \end{split} \end{equation*}$

where

$\begin{equation*} \begin{split} y _{0} (x)& =\frac{1}{2} + \frac{x}{2} + x^{2} + 3 x^{3} + 2 \ldots \\ y _{1} (x)& =-\frac{1}{2} + \frac{x}{2} - x^{2} + 3 x^{3} + 2 \ldots \end{split} \end{equation*}$

CHI.2.2.2 General form

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

The coefficients $u (n)$

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

$\begin{equation*} \begin{split} & -2u (n) n + u (n - 1) \bigl(-3 + 3 (n - 1)^{2} + 5 n\bigr) + \\ & u (n - 2) \bigl(8 - 5 n - 4 (n - 2)^{2} - (n - 2)^{3}\bigr)=0 \end{split} \end{equation*}$

whose initial conditions are given by

$\begin{equation*} \begin{split} u (1)& =\frac{1}{2} \\ u (0)& =\frac{1}{2} \end{split} \end{equation*}$

This recurrence has the closed form solution

$\begin{equation*} \begin{split} u (n)& =\frac{\Gamma (n + 1)}{2}. \end{split} \end{equation*}$

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

The coefficients $u (n)$

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

$\begin{equation*} \begin{split} & -2u (n) n + u (n - 1) \bigl(3 - 3 (n - 1)^{2} - 5 n\bigr) + \\ & u (n - 2) \bigl(8 - 5 n - 4 (n - 2)^{2} - (n - 2)^{3}\bigr)=0 \end{split} \end{equation*}$

whose initial conditions are given by

$\begin{equation*} \begin{split} u (1)& =\frac{1}{2} \\ u (0)& =-\frac{1}{2} \end{split} \end{equation*}$

This recurrence has the closed form solution

$\begin{equation*} \begin{split} u (n)& =\frac{-(-1)^{n} \Gamma (n + 1)}{2}. \end{split} \end{equation*}$