Proof of the error function being holomorphic and independent of path

I have been searching around and can’t seem to find any answer to the following two questions anywhere and it has started to become really frustrating!

1. Prove that the error function is holomorphic on $mathbb{C}$
2. Prove that the error function is independent of the path between the boundaries of the integral (How can I formulate this question better?)

For reference erf(z) = $operatorname{erf} z = frac{2}{sqrtpi}int_0^z e^{-t^2},dt.$

Thanks in advance!