# ERFI.1 Introduction

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

ERFI.1.1

The initial conditions of ERFI.1.1 are given at by

 ERFI.1.2

# ERFI.2 Series and asymptotic expansions

## ERFI.2.1 Taylor expansion at

ERFI.2.1.1.1

### ERFI.2.1.2 General form

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

## ERFI.2.2 Asymptotic expansion at

### ERFI.2.2.1 First terms

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where

### ERFI.2.2.2 General form

#### ERFI.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

#### ERFI.2.2.2.2 Auxiliary function

The auxiliary function has the exact form

# ERFI.3 Graphs

## ERFI.3.1 Real axis

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## ERFI.3.2 Complex plane

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