ERFC Complementary Error Function

ERFC.1 Introduction

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

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

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

erfc(0) = 1,
\frac{∂ erfc(x)}{∂x}(0) = \frac{−2}{\sqrt{π}} .

(ERFC.1.2)

Related function: Error Function

ERFC.2 Series and asymptotic expansions

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

\begin{aligned}erfc(x) & = 1 − \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} (ERFC.2.1.1.1)

ERFC.2.1.2 General form.

erfc(x) = ∑_{n=0}^{∞} u(n)x^{n}.

(ERFC.2.1.2.1)

The coefficients u(n) satisfy the recurrence

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

Initial conditions of ERFC.2.1.2.2 are given by

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

ERFC.2.2 Asymptotic expansion at ∞.

ERFC.2.2.1 First terms.

erfc(x) ≈ e^{(− \frac{1}{x^{2}} )} xy_{0}(x),

where

y_{0}(x) = π^{(− \frac{1}{2})} − \frac{x^{2}}{2\sqrt{π}} + 2 …

ERFC.2.2.2 General form. ERFC.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(1) & = 0 \\ u(0) & = π^{(− \frac{1}{2})} \end{aligned}

This recurrence has the closed form solution

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

ERFC.3 Graphs

ERFC.3.1 Real axis.

ERFC.3 GRAPHS 3

ERFC.3.2 Complex plane.