Can I use the asymptotic expansion of the Error function for complex arguments?

Question

I tried to evaluate a limit of the form $$ \lim_{x\rightarrow\infty} \mathrm{erfi}\left(zx\right) $$ for some complex number $z\neq0$. Following this answer, one idea would be to write $iz=\left|z\right|e^{i\mathrm{arg}(iz)}$ and then use the asmyptotic expansion of the error function at $\infty$. Is this a possible way of calculating these limits? Can I just plug in complex values into the asymptotic expansion? I don't know much about these expansions, for example how they are obtained.

Answer 1

Yes.

It is technically true that you cannot do this with arbitrary functions. HOWEVER, the error function is a "homomorphic" function, which when dealing with complex functions means that it is continuous and complex differentiable across the complex plane wherever it is finitely defined. When dealing with complex functions, a function is holomorphic if and only if it is also "analytic", meaning its Taylor Series expansions converge to the function when they converge. The error function is indeed analytic, as cited here.