I've been given the following equations; $\psi_1(z)=\frac{e^{z^2}}{\sqrt{\pi}}\int_{-\infty}^{z}e^{-t^2}dt$ (i.e. integrate over the straight line contour {$t + z: -\infty\lt t \lt 0 $}) $\psi_2(z)=\frac{e^{z^2}}{\sqrt{\pi}}\int_{\infty}^{z}e^{-t^2}dt$ (i.e. integrate over the straight line contour {$t + z: \infty\gt t \gt 0 $})
I am asked to show $\psi_1(z)$ is bounded for Re(z)$\leq$0 and similarly show $\psi_2(z)$ is bounded for Re(z)$\geq$0. Also I want to show the limit of $\psi_1(x)$ as $x\rightarrow -\infty$ and $\psi_2(x)$ as $x\rightarrow \infty$
What I have done so far is to show that these equations can be represented in terms of the error function, that is;
\begin{align} \psi_1(z)&=\frac{e^{z^2}}{2}(\operatorname{erf}(z)+1) \\ \psi_2(z)&=\frac{e^{z^2}}{2}(\operatorname{erf}(z)-1) \end{align}
Yet am having trouble finding an upper bound for the above functions.