SI Sine Integral

SI.1 Introduction

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

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

SI.1.1

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

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

SI.1.2

Related function: Cosine Integral

SI.2 Series and asymptotic expansions

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SI.2.1 Asymptotic expansion at

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SI.2.1.1 First terms

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

SI.2.1.1.1

SI.2.1.2 General form

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

SI.2.1.2.1

The coefficients $u (n)$

satisfy the recurrence

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

SI.2.1.2.2

Initial conditions of SI.2.1.2.2 are given by

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

SI.2.1.2.3

The recurrence SI.2.1.2.2 has the closed form solution

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

SI.2.1.2.4

SI.2.2 Asymptotic expansion at $\infty$

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SI.2.2.1 First terms

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

where

$\begin{equation*} \begin{split} y _{0} (x)& =terms _{\Bigl[1,1,\Bigl[\Bigl[0,\Bigl[\Bigl[0,\frac{\pi}{2}\Bigr]\Bigr]\Bigr]\Bigr]\Bigr]} + \ldots \\ y _{1} (x)& =-\frac{1}{2} - \frac{\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) x}{2} - \\ & \quad{}\quad{}-\left(-\frac{1}{4} - \frac{5 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}}{4}\right) x^{2} - -\Biggl(-\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) + \\ & \quad{}\quad{}4 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \left(-\frac{1}{4} - \frac{5 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}}{4}\right)\Biggr) x^{3} + 2 \ldots \\ y _{2} (x)& =-\frac{1}{2} - \frac{\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) x}{2} - \\ & \quad{}\quad{}-\left(-\frac{1}{4} - \frac{5 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}}{4}\right) x^{2} - -\Biggl(-\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) + \\ & \quad{}\quad{}4 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \left(-\frac{1}{4} - \frac{5 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}}{4}\right)\Biggr) x^{3} + 2 \ldots \end{split} \end{equation*}$

SI.2.2.2 General form

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

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

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

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

The coefficients $u (n)$

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

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

whose initial conditions are given by

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

SI.2.2.2.3 Auxiliary function $y _{2} (x)$

The coefficients $u (n)$

of $y _{2} (x)$ satisfy the following recurrence

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

whose initial conditions are given by

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