# ERF.1 Introduction

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

ERF.1.1

The initial conditions of ERF.1.1 are given at by

 ERF.1.2

Related function: Complementary Error Function

# ERF.2 Series and asymptotic expansions

## ERF.2.1 Taylor expansion at

ERF.2.1.1.1

### ERF.2.1.2 General form

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 ERF.2.1.2.1
The coefficients satisfy the recurrence
ERF.2.1.2.2
Initial conditions of ERF.2.1.2.2 are given by
ERF.2.1.2.3

## ERF.2.2 Asymptotic expansion at

### ERF.2.2.1 First terms

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where

### ERF.2.2.2 General form

#### ERF.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

#### ERF.2.2.2.2 Auxiliary function

The auxiliary function has the exact form

# ERF.3 Graphs

## ERF.3.1 Real axis

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## ERF.3.2 Complex plane

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