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SI Sine Integral

SI.1 Introduction

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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 $0$

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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*}
 
 
 
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