ERFI Imaginary Error Function

ERFI.1 Introduction

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

\begin{aligned}& −2x\frac{∂y(x)}{∂x} + \frac{∂^{2}y(x)}{∂x^{2}} & = 0. \\ (ERFI.1.1) \end{aligned}

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

\begin{aligned}erfi(0) & = 0, \\ \frac{∂ erfi(x)}{∂x}(0) & = \frac{2}{\sqrt{π}} . \end{aligned} (ERFI.1.2)

ERFI.2 Series and asymptotic expansions

ERFI.2.1 Taylor expansion at 0. ERFI.2.1.1 First terms.

\begin{aligned}erfi(x) & = \frac{2}{\sqrt{π}} x + \frac{2}{3\sqrt{π}} x^{3} + \frac{1}{5\sqrt{π}} x^{5} + \frac{1}{21\sqrt{π}} x^{7} + \frac{1}{108\sqrt{π}} x^{9}+ \\ & \frac{1}{660\sqrt{π}} x^{11} + \frac{1}{4680\sqrt{π}} x^{13} + \frac{1}{37800\sqrt{π}} x^{15} + O(x^{16}). \end{aligned} (ERFI.2.1.1.1)

ERFI.2.1.2 General form.

\begin{aligned}& erfi(x) & = ∑_{n=0}^{∞} u(n)x^{n}. \\ (ERFI.2.1.2.1) \end{aligned}

The coefficients u(n) satisfy the recurrence

\begin{aligned}& −2nu(n) + (n^{2} + 3n + 2)u(n + 2) & = 0. \\ (ERFI.2.1.2.2) \end{aligned}

Initial conditions of ERFI.2.1.2.2 are given by

\begin{aligned}u(0) & = 0, \\ u(1) & = \frac{2}{\sqrt{π}} . \end{aligned} ; (ERFI.2.1.2.3)

ERFI.2.2 Asymptotic expansion at ∞.

ERFI.2.2.1 First terms.

erfi(x) ≈ e^{x^{(−2)}} xy_{0}(x) + ser_{[1,1,[[0,[[0,−i]]]]]},

where

\begin{aligned}y_{0}(x) & = π^{(− \frac{1}{2})} + \frac{x^{2}}{2\sqrt{π}} + 2 … \\ y_{1}(x) & = terms_{[1,1,[[0,[[0,−i]]]]]} + … \end{aligned}

ERFI.2.2.2 General form. ERFI.2.2.2.1 Auxiliary function y_{0}(x). The coefficients u(n) of y_{0}(x) satisfy the

following recurrence

−2nu(n) + u(n − 2)(−4 + 3n + (n − 2)^{2}) = 0

whose initial conditions are given by

\begin{aligned}u(0) & = π^{(− \frac{1}{2})} \\ u(1) & = 0 \end{aligned}

This recurrence has the closed form solution

\begin{aligned}u(2n) & = \frac{Γ(n + \frac{1}{2})}{π}, \\ u(2n + 1) & = 0. \end{aligned}

ERFI.2.2.2.2 Auxiliary function y_{1}(x). The auxiliary function y_{1}(x) has the exact form

y_{1}(x) = −i

ERFI.3 Graphs

ERFI.3.1 Real axis.

ERFI.3 GRAPHS 3

ERFI.3.2 Complex plane.