Wikipedia has the following definition for imaginary error function:
$$\textrm{erfi}(x) = i ~\textrm{eft}(ix) $$
I have two questions,
- Is the following correct?
$$\textrm{erfi}(ix) = i ~\textrm{eft}(i^2 x) $$
- What about this one? Where to start in simplfying this?
$$\textrm{erfi}(i\sqrt{i} x) = ? $$
The definition you provide from Wikipedia is misleading, and possibly inaccurate. More appropriately stated, the error function of a purely imaginary number, say $z=iy$ is given by
$$\text{erf}(iy)=\frac{2i}{\sqrt{\pi}}\int_0^y e^{t^2}=\frac{2i}{\sqrt{\pi}}e^{y^2}D(y)$$
Reference: K. Oldham, J. Myland, & J. Spanier, An Atlas of Functions, Springer. Chapter 7 covers the error function.
The two questions you pose must be answered in the context of these definitions.
In my work with the error function of complex arguments, I have programmed up the equations given in Abramowitz, M., Stegun, I.A., 1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, Equation 7.1.29.