SHI Hyperbolic Sine Integral

SHI.1 Introduction

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

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

SHI.1.1

Although $0$ is a singularity of SHI.1.1, the initial conditions can be given by

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

SHI.1.2

Related function: Hyperbolic Cosine Integral

SHI.2 Series and asymptotic expansions

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SHI.2.1 Asymptotic expansion at $\infty$

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SHI.2.1.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{Shi} (x)\approx \\ & \quad{}\quad{}ser _{\Bigl[1,1,\Bigl[\Bigl[0,\Bigl[\Bigl[0,\frac{\pi}{2}\Bigr]\Bigr]\Bigr]\Bigr]\Bigr]} + \operatorname{e} ^{\frac{1}{x}} x y _{1} (x) + \operatorname{e} ^{\Bigl(-\frac{1}{x}\Bigr)} x y _{2} (x), \end{split} \end{equation*}$

where

$\begin{equation*} \begin{split} y _{0} (x)& =terms _{\Bigl[1,1,\Bigl[\Bigl[0,\Bigl[\Bigl[0,\frac{\pi}{2}\Bigr]\Bigr]\Bigr]\Bigr]\Bigr]} + \ldots \\ y _{1} (x)& =\frac{1}{2} + \frac{x}{2} + x^{2} + 3 x^{3} + 2 \ldots \\ y _{2} (x)& =\frac{1}{2} - \frac{x}{2} + x^{2} - 3 x^{3} + 2 \ldots \end{split} \end{equation*}$

SHI.2.1.2 General form

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SHI.2.1.2.1 Auxiliary function $y _{0} (x)$

The auxiliary function $y _{0} (x)$ has the exact form

$\begin{equation*} \begin{split} y _{0} (x)& =\frac{\pi}{2} \end{split} \end{equation*}$

SHI.2.1.2.2 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

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

SHI.2.1.2.3 Auxiliary function $y _{2} (x)$

The coefficients $u (n)$

of $y _{2} (x)$ 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

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

This recurrence has the closed form solution

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

SHI.2.2 Asymptotic expansion at

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SHI.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{Shi} (x)\approx x \Biggl(1 + \frac{x^{2}}{18} + \frac{x^{4}}{600} + \frac{x^{6}}{35280} + \frac{x^{8}}{3265920} + \frac{x^{10}}{439084800} + \frac{x^{12}}{80951270400} + \\ & \quad{}\quad{}\frac{x^{14}}{19615115520000}\ldots\Biggr). \end{split} \end{equation*}$

SHI.2.2.1.1

SHI.2.2.2 General form

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$\begin{equation*} \begin{split} & \operatorname{Shi} (x)\approx x \sum_{n = 0}^{\infty} u (n) x^{n}. \end{split} \end{equation*}$

SHI.2.2.2.1

The coefficients $u (n)$

satisfy the recurrence

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

SHI.2.2.2.2

Initial conditions of SHI.2.2.2.2 are given by

$\begin{equation*} \begin{split} u (1)& =0, \\ u (0)& =1, \\ u (2)& =\frac{1}{18}. \end{split} \end{equation*}$

SHI.2.2.2.3

The recurrence SHI.2.2.2.2 has the closed form solution

$\begin{equation*} \begin{split} u (2 n + 1)& =0, \\ u (2 n)& =\frac{1}{(2 n + 1) \Gamma (2 n + 2)}. \end{split} \end{equation*}$

SHI.2.2.2.4