We consider rational approximations to the Faddeeva or plasma dispersion function, defined
as
$w(z) = e^{-z^{2}} \mbox{erfc} (-iz)$.
With many important applications in physics, good software for
computing the function reliably everywhere in the complex plane is required. In this talk
we shall derive rational approximations to $w(z)$ via quadrature, M\"{o}bius transformations, and best approximation. The various approximations are compared with regard to speed of convergence, numerical stability, and ease of generation of the coefficients of the formula.
In addition, we give preference to methods for which a single expression yields uniformly
high accuracy in the entire complex plane, as well as being able to reproduce exactly the
asymptotic behaviour
$w(z) \sim i/(\sqrt{\pi} z), z \rightarrow \infty$
(in an appropriate sector).
This is Joint work with: Stephan Gessner, St\'efan van der Walt