# Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function:

$$\mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right)$$

$z \in \Bbb{C}$

which also has the following integral representation: $$\mathrm{erfcx}(z) = -{\frac{i}{\sqrt{\pi}}}\int {\frac{e^{-t^2}}{t-iz} dt}$$

1. What is the correct decomposition into real and imaginary parts? Are they related to other known functions?
2. What are the symmetries of this function?
3. Estimate $\mathrm{erfcx}(z) - \mathrm{erfcx}(z^*)$ ($z^*$ denoting complex conjugation)
4. Does $\mathrm{erfcx}(z) - \mathrm{erfcx}(z^*)$ have any notable properties? What are its stationary points (e.g. along lines parallel to the real axis)?
5. Can anyone suggest possible routes for graphing the real and imaginary parts of this function?

Motivation

The complementary error function appears in the solution of a transport problem that I am trying to solve - I have exposed it to some detail here. I hope that the study of this function will enable the suitability of my solution for the required application.

Here are my attempts to solve these questions - I believe I have made some progress, but they are incomplete.

## 1.

It is worth noting that $\mathrm{erfcx}(z)=w(iz)$ where $w(z)$ is the Faddeeva function. Therefore, the properties of $\mathrm{erfcx}(z)$ should follow trivially from that relation (as they are given, for example in Abramowitz & Stegun). However, I do find in the literature the restriction of many of the properties to the upper half of the imaginary plane, but as I would like to see if such restrictions can be avoided I shall state the more salient ones explicitly.

Let $z = \alpha + i \beta$; $\alpha , \beta \in \Bbb{R}$. Then

$$\mathrm{erfcx}(z) = u(\alpha,\beta)+ i\ v(\alpha,\beta)=$$ $$e^{\alpha^2 -\beta^2} \left[ \cos{(2\alpha\beta)}(1 - \Re[\mathrm{erf}z]) + \sin{(2\alpha\beta)}\Im[\mathrm{erf}z] \right] -i\ e^{\alpha^2 -\beta^2} \left[ \cos{(2\alpha\beta)}\Im[\mathrm{erf}z] + \sin{(2\alpha\beta)}(1 - \Re[\mathrm{erf}z]) \right]$$

which (taking $\Re[\mathrm{erf}z]$ to be odd wrt $\alpha$ and even wrt $\beta$ and conversely $\Im[\mathrm{erf}z]$ even wrt $\alpha$ and odd wrt $\beta$ - if I am not terribly mistaken) implies $u(\alpha,\beta)=u(\alpha,-\beta)$ and $v(\alpha,\beta)=-v(\alpha,-\beta)$.

As the Faddeeva function $w(z)$ is decomposed to (real and imaginary) Voigt functions

$$w(p + iq) = U(p,q)+i\ V(p,q)$$

one is tempted to write $u(\alpha,\beta)=U(-\beta, \alpha)$ and $v(\alpha,\beta)=V(-\beta, \alpha)$

But does this relation hold $\forall \alpha , \beta \in \Bbb{R}$? Moreover, are there calculation methods $v$ and $u$ and relations to other commonly used special functions? With appropriate scaling, for $\alpha>0$ $U$ is related to the Voigt profile; for $\alpha=0$, $V$ is related to Dawson's integral $\sqrt{\pi/4}{e^{-x^2}\mathrm{erfi}(x)}$. But can a more generalised representation be found, valid for all $z$?

## 2.

$u$ has even parity wrt $\beta$ and $v$ is odd wrt $\beta$. The symmetries of $w(z)$ would be expected to hold for $\mathrm{erfcx}(z)$ as well.

## 3.

$$\mathrm{erfcx}(z) - \mathrm{erfcx}(z^*) = 2 i \ v = -2i\ e^{\alpha^2 -\beta^2} \left( \cos{(2\alpha\beta}\Im[\mathrm{erf}z] + \sin{2\alpha\beta}(1- \Re[\mathrm{erf}z]) \right)$$

showing that this difference is purely imaginary. But a way to calculate this wouldbe useful.

## 4.

I have not yet looked into this problem in any particular detail; the ODE representation of $w$ and the associated recurrence relations will probably be of use here.

## 5.

I have found a number of C libraries for complex error functions (e.g. here , which includes a short bibliography for the calculation). I am in the process of implementing them; but if there are other quidirty hacks for estimating the imaginary and real parts of $\mathrm{erfcx}(z)$ I would be very glad to hear about them.

As the expressions I obtain for $\Re[\mathrm{erf}(\alpha + i \beta)]$ and $\Im[\mathrm{erf}(\alpha + i \beta)]$ are long-winded (and often encountered in the relevant literature) I shall append them here.
(a) Migrating along the real axis to $z=\alpha$ and then up to $z=\alpha + i\beta$ $$\mathrm{erf}(\alpha + i \beta) = \sqrt{\frac{4}{\pi}}\int _{0}^{\alpha}{e^{-t^2} dt} + i \sqrt{\frac{4}{\pi}}\int _{0}^{\beta}{e^{-(\alpha+ i \ t)^2} dt} =$$ $$\mathrm{erf}(\alpha) + \sqrt{\frac{4}{\pi}} e^{-\alpha^2} \int _{0}^{\beta}{e^{t^2} \sin{(2\alpha t)} dt} + i \sqrt{\frac{4}{\pi}} e^{-\alpha^2} \int _{0}^{\beta}{e^{t^2} \cos{(2\alpha t)} dt}$$ (b) Migrating along the imaginary axis to $z=i\beta$ and then parallel to the real axis to $z=\alpha + i\beta$ $$\mathrm{erf}(\alpha + i \beta) = i\sqrt{\frac{4}{\pi}}\int _{0}^{\beta}{e^{t^2} dt} + \sqrt{\frac{4}{\pi}}\int _{0}^{\alpha}{e^{-(t + i \beta)^2} dt} =$$ $$\sqrt{\frac{4}{\pi}} e^{\beta^2} \int _{0}^{\alpha}{e^{-t^2} \cos{2 \beta t} dt} + i\sqrt{\frac{4}{\pi}} e^{\beta^2} \int _{0}^{\alpha}{e^{-t^2} \sin{2 \beta t} dt} + i\ \mathrm{erfi}(\beta)$$
I find the occurrence of definite integrals of the form$\int _{0}^{\kappa}{e^{\pm t^2} \cos{2 \lambda t} dt}$ and $\int _{0}^{\kappa}{e^{\pm t^2} \sin{2 \lambda t} dt}$ quite suggestive, esp. when seen e.g. in conjunction with an integral representation of Dawson’s function:
$$D_{+} (\chi)= \int_{0}^{\infty}{e^{-t^2} \sin{2 \chi t} dt}$$