# HACS.1 Introduction

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

HACS.1.1

The initial conditions of HACS.1.1 are given at by

 HACS.1.2

Related functions: Inverse Cosine,Inverse Sine

# HACS.2 Series and asymptotic expansions

## HACS.2.1 Asymptotic expansion at

### HACS.2.1.1 First terms

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 HACS.2.1.1.1

### HACS.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).

## HACS.2.2 Asymptotic expansion at

HACS.2.2.1.1

### HACS.2.2.2 General form

HACS.2.2.2.1
The coefficients satisfy the recurrence
HACS.2.2.2.2
Initial conditions of HACS.2.2.2.2 are given by
HACS.2.2.2.3
The recurrence HACS.2.2.2.2 has the closed form solution
HACS.2.2.2.4

## HACS.2.3 Asymptotic expansion at

### HACS.2.3.1 First terms

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 HACS.2.3.1.1

### HACS.2.3.2 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).

## HACS.2.4 Taylor expansion at

HACS.2.4.1.1

### HACS.2.4.2 General form

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 HACS.2.4.2.1
The coefficients satisfy the recurrence
HACS.2.4.2.2
Initial conditions of HACS.2.4.2.2 are given by
HACS.2.4.2.3

# HACS.3 Graphs

## HACS.3.1 Real axis

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## HACS.3.2 Complex plane

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