SHI Hyperbolic Sine Integral

SHI.1 Introduction

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

\begin{aligned}& −x\frac{∂y(x)}{∂x} + 2\frac{∂^{2}y(x)}{∂x^{2}} + x\frac{∂^{3}y(x)}{∂x^{3}} & = 0. \\ (SHI.1.1) \end{aligned}

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

\begin{aligned}& \frac{∂ \frac{Shi(x)}{x}}{∂x} & = 1. \\ (SHI.1.2) \end{aligned}

Related function: Hyperbolic Cosine Integral

SHI.2 Series and asymptotic expansions

SHI.2.1 Asymptotic expansion at ∞. SHI.2.1.1 First terms.

\begin{matrix}Shi(x) ≈ \\ ser_{[1,1,[[0,[[0, \frac{π}{2} ]]]]]} + e^{\frac{1}{x}} xy_{1}(x) + e^{(− \frac{1}{x})} xy_{2}(x), \end{matrix}

where

\begin{aligned}y_{0}(x) & = terms_{[1,1,[[0,[[0, \frac{π}{2} ]]]]]} + … \\ y_{1}(x) & = \frac{1}{2} + \frac{x}{2} + x^{2} + 3x^{3} + 2 … \\ y_{2}(x) & = \frac{1}{2} − \frac{x}{2} + x^{2} − 3x^{3} + 2 … \end{aligned}

SHI.2.1.2 General form. SHI.2.1.2.1 Auxiliary function y_{0}(x). The auxiliary function y_{0}(x) has the exact

form

y_{0}(x) = \frac{π}{2}

SHI.2.1.2.2 Auxiliary function y_{1}(x). The coefficients u(n) of y_{1}(x) satisfy the following recurrence

\begin{matrix}− 2u(n)n + u(n − 1)(−3 + 3(n − 1)^{2} + 5n)+ \\ u(n − 2)(8 − 5n − 4(n − 2)^{2} − (n − 2)^{3}) = 0 \end{matrix}

whose initial conditions are given by

\begin{aligned}u(1) & = \frac{1}{2} \\ u(0) & = \frac{1}{2} \end{aligned}

This recurrence has the closed form solution

u(n) = \frac{Γ(n + 1)}{2}.

SHI.2.1.2.3 Auxiliary function y_{2}(x). The coefficients u(n) of y_{2}(x) satisfy the following recurrence

\begin{matrix}− 2u(n)n + u(n − 1)(3 − 3(n − 1)^{2} − 5n)+ \\ u(n − 2)(8 − 5n − 4(n − 2)^{2} − (n − 2)^{3}) = 0 \end{matrix}

whose initial conditions are given by

\begin{aligned}u(0) & = \frac{1}{2} \\ u(1) & = −\frac{1}{2} \end{aligned}

This recurrence has the closed form solution

u(n) = \frac{(−1)^{n}Γ(n + 1)}{2}.

SHI.2.2 Asymptotic expansion at 0.

SHI.2.2.1 First terms.

\begin{matrix}Shi(x) ≈ x(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}+ \\ \frac{x^{14}}{19615115520000} …). \end{matrix} ; (SHI.2.2.1.1)

SHI.2.2.2 General form.

\begin{aligned}& Shi(x) ≈ x ∑_{n=0}^{∞} u(n)x^{n}. \\ (SHI.2.2.2.1) \end{aligned}

The coefficients u(n) satisfy the recurrence

\begin{aligned}& −u(n)(n + 1)(1 + n − (n + 1)^{2}) + u(n − 2)(1 − n) & = 0. \\ (SHI.2.2.2.2) \end{aligned}

Initial conditions of SHI.2.2.2.2 are given by

\begin{aligned}u(1) & = 0, \\ u(0) & = 1, \\ u(2) & = \frac{1}{18}. \end{aligned} ; (SHI.2.2.2.3)

The recurrence SHI.2.2.2.2 has the closed form solution

\begin{aligned}u(2n + 1) & = 0, \\ u(2n) & = \frac{1}{(2n + 1)Γ(2n + 2)}. \end{aligned} ; (SHI.2.2.2.4)