CHI Hyperbolic Cosine Integral

CHI.1 Introduction

Let x be a complex variable of ℂ \ {0, ∞}.The function Hyperbolic Cosine Integral (noted Chi) 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. \\ (CHI.1.1) \end{aligned}

The initial conditions of CHI.1.1 at 0 are not simple to state, since 0 is a (regular) singular point.

Related function: Hyperbolic Sine Integral

CHI.2 Series and asymptotic expansions

CHI.2.1 Asymptotic expansion at 0. CHI.2.1.1 First terms.

Chi(x) ≈ (\frac{−x^{2}}{4} − \frac{x^{4}}{96} − \frac{x^{6}}{4320} − \frac{x^{8}}{322560} − ln(x) + γ …). (CHI.2.1.1.1)

CHI.2.1.2 General form. The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).

CHI.2.2 Asymptotic expansion at ∞. CHI.2.2.1 First terms.

Chi(x) ≈ e^{\frac{1}{x}} xy_{0}(x) + e^{(− \frac{1}{x})} xy_{1}(x),

where

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

CHI.2.2.2 General form. CHI.2.2.2.1 Auxiliary function y_{0}(x). The coefficients u(n) of y_{0}(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}

2 CHI HYPERBOLIC COSINE INTEGRAL

This recurrence has the closed form solution

u(n) = \frac{Γ(n + 1)}{2}.

CHI.2.2.2.2 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{−(−1)^{n}Γ(n + 1)}{2}.