SI.1 Introduction

Let be a complex variable of .The function Sine Integral (noted ) is defined by the following third order differential equation

SI.1.1

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

 SI.1.2

Related function: Cosine Integral

SI.2 Series and asymptotic expansions

SI.2.1 Asymptotic expansion at

SI.2.1.1.1

SI.2.1.2 General form

SI.2.1.2.1
The coefficients satisfy the recurrence
SI.2.1.2.2
Initial conditions of SI.2.1.2.2 are given by
SI.2.1.2.3
The recurrence SI.2.1.2.2 has the closed form solution
SI.2.1.2.4

SI.2.2 Asymptotic expansion at

SI.2.2.1 First terms

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where

SI.2.2.2 General form

SI.2.2.2.1 Auxiliary function

The auxiliary function has the exact form

SI.2.2.2.2 Auxiliary function

The coefficients of satisfy the following recurrence whose initial conditions are given by

SI.2.2.2.3 Auxiliary function

The coefficients of satisfy the following recurrence whose initial conditions are given by

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