**Erfi imaginary error function**

## Introduction

Let x be a complex variable of C\backslash\;{\infty} .The function Imaginary Error Function

(noted erfi) is defined by the following second-order differential equation

-2x\;\frac{\partial y(x)}{\partial x}+\frac{\partial^2y(x)}{\partial x^2}=0———-(1)

The initial condition of erfi (1) at 0 is

erfi(0)=0

\;\frac{\partial erfi\;(x)}{\partial x}(0)={\textstyle\frac2{\sqrt\pi}}

## ERFI.2 Series and Asymptotic Expansions

**ERFI.2.1 Taylor expansion at 0.**

**ERFI.2.1.1 First terms.**

erfi\;(x)=\frac2{\sqrt\pi}x+\frac2{3\sqrt\pi}x^3+\frac1{5\sqrt\pi}x^5+\frac1{21\sqrt\pi}x^7+\frac1{108\sqrt\pi}x^9+\frac1{660\sqrt\pi}x^{11}+\frac1{4680\sqrt\pi}x^{13}+\frac1{37800\sqrt\pi}x^{15}+O(x^{16})

**ERFI.2.1.2 General form.**

erfi(x)=\sum_{n=0}^\infty u(n)x^n

The coefficients u(n) satisfy the recurrence

-2nu(n)+(n2 + 3n+2)u(n+2)=0 —————(ERFI.2.1.2.2)

Initial conditions of **ERFI.2.1.2.2** are given by

U(0)=0

U(1)=\frac2{\sqrt\pi}