ACS.1 Introduction

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

ACS.1.1

The initial conditions of ACS.1.1 are given at by

 ACS.1.2

Related functions: Inverse Sine,Inverse Hyperbolic Cosine

ACS.2 Series and asymptotic expansions

ACS.2.1 Asymptotic expansion at

ACS.2.1.1 First terms

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 ACS.2.1.1.1

ACS.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).

ACS.2.2 Asymptotic expansion at

ACS.2.2.1.1

ACS.2.2.2 General form

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

ACS.2.3 Asymptotic expansion at

ACS.2.3.1 First terms

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 ACS.2.3.1.1

ACS.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).

ACS.2.4 Taylor expansion at

ACS.2.4.1.1

ACS.2.4.2 General form

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

ACS.3 Graphs

ACS.3.1 Real axis

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ACS.3.2 Complex plane

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