In what ways can I extend the error function to accept complex arguments?

Question

What are the different approaches to extending the error function to accept complex arguments? When should I favor using one approach over another?

Answer 1

Since $\mathrm{erf}$ is analytic/holomorphic over the real line (in particular, you have a power series representation that converges for all reals), you have the deep result of (complex) analysis that there is a unique analytic continuation of $\mathrm{erf}$ to the entire complex plane.

Thus,

In what ways can I extend the error function to accept complex arguments?

• Only one, good sir.

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As for extending algorithms to compute the error function for real arguments to complex arguments, bear in mind that although $\mathrm{erf}$ is relatively tame in the real line, it grows exponentially in other sectors of the complex plane, so special methods are needed for computing the complex error function. Often, however, in situations that involve computing $\mathrm{erf}$ for complex arguments, the much more application-relevant function is the Faddeeva/Voigt function, $w(z)=\exp(-z^2)(1-\mathrm{erf}(-iz))$; the references given in njuffa's comment point to some of the common methods for computing it.