Complex Error Function in Mathematica

Question

The complex error function w(z) is defined as e^(-x^2) erfc(-ix). The problem with using w(z) as defined above is that the erfc tends to explode out for larger x (complemented by the exponential going to 0 so everything stays small), so that Mathematica reverts to arbitrary precision calculations that make life VERY slow. The function is used in implementing the voigt profile - a line shape commonly used in spectroscopy and other related areas. Right now I'm reverting to calculating the lineshape once and using an interpolation to speed things up, however this doesn't let me alter the parameters of the lineshape (or fit to them) easily.

scipy has a nice and fast implementation of w(z) as scipy.special.wofz, and I was wondering if there is an equivalent in Mathematica.

Answer 1

The complex error function can be written in terms of the Hermite "polynomial" H_{-1}(x):

In[1]:= FullSimplify[2 HermiteH[-1,I x] == Sqrt[Pi] Exp[-x^2] Erfc[I x]]
Out[1]= True

And the evaluation does not suffer as many underflows and overflows

In[68]:= 2 HermiteH[-1, I x] /. x -> 100000.
Out[68]= 6.12323*10^-22 - 0.00001 I

In[69]:= Sqrt[Pi] E^-x^2 Erfc[I x] /. x -> 100000.
During evaluation of In[69]:= General::unfl: Underflow occurred in computation. >>
During evaluation of In[69]:= General::ovfl: Overflow occurred in computation. >>
Out[69]= Indeterminate

That said, some quick tests show that the evaluation speed of the Hermite function to be slower than that of the product of the exponential and error function...