# Asymptotic form of imaginary error function

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?