I am interested in the asymptotic form of the imaginary error function for large, real arguments. I find [1] the following:

$$ \text{erfi}(z) = -i + \frac{e^{z^2}}{\sqrt\pi}\left(z^{-1} + \frac12 z^{-3} + \dots \right) $$

How is that possible? Set $z = x \rightarrow \infty$ (where $x \in \mathbb{R}$) which gives at leading order

$$ \text{erfi}(x) \approx -i + \frac{e^{x^2}}{\sqrt{\pi}x} $$

I read: left-hand side = real number, right-hand side = complex number. How can that be correct?

[1] http://mathworld.wolfram.com/Erfi.html


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