ERFI Imaginary Error Function

ERFI.1 Introduction

Let $x$ be a complex variable of $\mathbb{C} \setminus \{\infty\}$ .The function Imaginary Error Function (noted $\operatorname{erfi}$ ) is defined by the following second order differential equation

$\begin{equation*} \begin{split} -2x \frac{\partial y (x)}{\partial x} + \frac{\partial^{2} y (x)}{\partial x^{2}}& =0. \end{split} \end{equation*}$

ERFI.1.1

The initial conditions of ERFI.1.1 are given at $0$ by

$\begin{equation*} \begin{split} \operatorname{erfi} (0)& =0, \\ \frac{\partial \operatorname{erfi} (x)}{\partial x} (0)& =\frac{2}{\sqrt{\pi}}. \end{split} \end{equation*}$

ERFI.1.2

ERFI.2 Series and asymptotic expansions

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ERFI.2.1 Taylor expansion at

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$\begin{equation*} \begin{split} \operatorname{erfi} (x)& =\frac{2}{\sqrt{\pi}} x + \frac{2}{3 \sqrt{\pi}} x^{3} + \frac{1}{5 \sqrt{\pi}} x^{5} + \frac{1}{21 \sqrt{\pi}} x^{7} + \frac{1}{108 \sqrt{\pi}} x^{9} + \\ & \quad{}\quad{}\frac{1}{660 \sqrt{\pi}} x^{11} + \frac{1}{4680 \sqrt{\pi}} x^{13} + \frac{1}{37800 \sqrt{\pi}} x^{15} + \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$

ERFI.2.1.1.1

ERFI.2.1.2 General form

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$\begin{equation*} \begin{split} \operatorname{erfi} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}. \end{split} \end{equation*}$

ERFI.2.1.2.1

The coefficients $u (n)$

satisfy the recurrence

$\begin{equation*} \begin{split} -2n u (n) + \bigl(n^{2} + 3 n + 2\bigr) u (n + 2)& =0. \end{split} \end{equation*}$

ERFI.2.1.2.2

Initial conditions of ERFI.2.1.2.2 are given by

$\begin{equation*} \begin{split} u (0)& =0, \\ u (1)& =\frac{2}{\sqrt{\pi}}. \end{split} \end{equation*}$

ERFI.2.1.2.3

ERFI.2.2 Asymptotic expansion at $\infty$

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ERFI.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{erfi} (x)\approx \operatorname{e} ^{x^{(-2)}} x y _{0} (x) + ser _{\bigl[1,1,\bigl[\bigl[0,\bigl[[0,-i]\bigr]\bigr]\bigr]\bigr]}, \end{split} \end{equation*}$

where

$\begin{equation*} \begin{split} y _{0} (x)& =\pi^{\Bigl(-\frac{1}{2}\Bigr)} + \frac{x^{2}}{2 \sqrt{\pi}} + 2 \ldots \\ y _{1} (x)& =terms _{\bigl[1,1,\bigl[\bigl[0,\bigl[[0,-i]\bigr]\bigr]\bigr]\bigr]} + \ldots \end{split} \end{equation*}$