I found the following relation from a 1968 paper:
$ \DeclareMathOperator\erf{erf} \int_{-\infty}^{\infty}\erf(x)\exp^{-(ax+b)^2}dx=-\frac{\sqrt\pi}{a}\erf\big(\frac{b}{\sqrt{a^2+1}}\big), Re(a^2)>0, $
where $\erf(x)$ is the error function, and $a$ and $b$ constants that do not depend on $x$.
Alternatively, here's a screenshot of relevant equation.
My question is: is this valid complex constants $a$ and $b$? The condition on $a$ suggests so, but nothing is said about $b$.
Furthermore, to put the integral I really wish to solve into the form above, I made the definition $t + z \equiv x$, where $t$ is real but $z$ is complex. Is that a legitimate move? I worry because $\erf(z)$ for complex $z$ blows up.
Thank you in advance.