# CHI.1 Introduction

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

CHI.1.1

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

Related function: Hyperbolic Sine Integral

# CHI.2 Series and asymptotic expansions

## CHI.2.1 Asymptotic expansion at

### CHI.2.1.1 First terms

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 CHI.2.1.1.1

### CHI.2.1.2 General form

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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

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where

### CHI.2.2.2 General form

#### CHI.2.2.2.1 Auxiliary function

The coefficients of satisfy the following recurrence whose initial conditions are given by This recurrence has the closed form solution

#### CHI.2.2.2.2 Auxiliary function

The coefficients of satisfy the following recurrence whose initial conditions are given by This recurrence has the closed form solution

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