I have the following equation to solve $$ z+e^{z^2}\operatorname{erfc}(z)=0 $$ being $$ \operatorname{erfc}(z)=1-\frac{2}{\sqrt{\pi}}\int_0^ze^{-s^2}ds. $$ I solved it numerically and appears to have an infinity of complex solutions. I am in need to find an approximate analytical approximation for the first root. I made several attempts, e.g. using either Taylor or asymptotic expansions, but to no avail. I was not able to recover the first root that seems to be $-.3378723369\ldots-.9499598705\ldots i$.
My question: is it possible to find a successful approximation scheme to get an analytical expression for the roots of the given equation?
Edit: There appear to be infinite roots in $\mathbb{C}$. This can be seen straightforwardly in Fig. 1 where both the real and imaginary part are given of $-z$ and $e^{z^2}\operatorname{erfc}(z)$
Orange is $Re(-z)$, green is $Im(-z)$. There is a common line between this twos crossing the part with the erfc showing a possible infinite set of roots.
I was able to find several roots by choosing a different starting point for the search algorithm. E.g. I have got, listing just a few along with their complex conjugates implied,
$-1.90593 - 1.82833 i$
$-3.19368 - 3.06637 i$
$-4.07604 - 3.94595 i$
$-6.71931 - 6.60297 i$
$-8.00551 - 7.89692 i$
$-10.2496 - 10.1527 i$
The aforementioned first root appears a recurring one in the search.
You can obtain the first root using Newton method.
Using for example $z_0=1+i$, it would give the following iterates $$\left( \begin{array}{cc} n & z_n \\ 0 & 1+ i \\ 1 & -0.570752+0.455687 i \\ 2 & -0.217277+0.691623 i \\ 3 & -0.298698+1.012267 i \\ 4 & -0.336117+0.946645 i \\ 5 & -0.337868+0.949970 i \\ 6 & -0.337872+0.949960 i \end{array} \right)$$
Using $z_0=1-i$, it would give the following iterates $$\left( \begin{array}{cc} n & z_n \\ 0 & 1-i \\ 1 & -0.570752-0.455687 i \\ 2 & -0.217277-0.691623 i \\ 3 & -0.298698-1.012267 i \\ 4 & -0.336117-0.946645 i \\ 5 & -0.337867-0.949968 i \\ 6 & -0.337872-0.949960 i \end{array} \right)$$
Starting with $z_0=i$ seems to converge faster $$\left( \begin{array}{cc} n & z_n \\ 0 & i \\ 1 & -0.400169+0.909374 i \\ 2 & -0.339614+0.946181 i \\ 3 & -0.337864+0.949950 i \\ 4 & -0.337872+0.949960 i \end{array} \right)$$
The same starting with $z_0=-i$ $$\left( \begin{array}{cc} n & z_n \\ 0 & - i \\ 1 & -0.400169-0.909374 i \\ 2 & -0.339614-0.946181 i \\ 3 & -0.337864-0.949950 i \\ 4 & -0.337872-0.949960 i \end{array} \right)$$
Concerning approximations, I am stuck. The only thing I really tried is a series expansion around $z=i$ and got $$\frac{-i \text{erfi}(1)+1+i e}{e}+\frac{\left(2 \text{erfi}(1)+2 i+e-\frac{2 e}{\sqrt{\pi }}\right) (z-i)}{e}+\frac{\left(i \text{erfi}(1)-1-\frac{2 i e}{\sqrt{\pi }}\right) (z-i)^2}{e}+O\left((z-i)^3\right)$$ Ignoring the higher order terms, solving the quadratic in $(z-i)$ and evaluating leads to $$z\approx -0.351584 + 0.947053 i$$ Doing the same around $z=-i$ gives the other root.
Edit
In terms of approximation of the solution, we can do better using the simplest $[1,1]$ Padé approximant built at $z_\pm=\pm i$. For example, $$z_+=-\frac{i \left(-4 e \left(\pi -2 \sqrt{\pi }\right) (\text{erfi}(1)+i)-5 \pi (\text{erfi}(1)+i)^2+4 e^2 \left(\sqrt{\pi }-1\right)\right)}{e \left(5 \pi -6 \sqrt{\pi }\right) (\text{erfi}(1)+i)+3 \pi (\text{erfi}(1)+i)^2+e^2 \left(4-6 \sqrt{\pi }+\pi \right)}$$ which is $-0.347274 + 0.955038 i$ much better than the previous one.
We could continue with $[1,n]$ Padé approximant to get explicit approximations (the expressions are really messy). The values are reporteds below $$\left( \begin{array}{cc} n & z^{(n)}_+\\ 0 & -0.400169+0.909374 i \\ 1 & -0.347274+0.955038 i \\ 2 & -0.337395+0.951302 i \\ 3 & -0.337591+0.949901 i \\ 4 & -0.337859+0.949907 i \end{array} \right)$$
Update
Looking at the table of the roots given in Jon' edit, they really look like $z_k \sim -k \pm ki$. Using the $[1,1]$ Padé approximant built at $z=-k-ik$ (the formula is too messy to be reported here) an approximation of the root can be obtained.
Below are listed some values (to be compared to Jon's) $$\left( \begin{array}{cc} k & \text{approximation} \\ 1 & -0.34995-0.95803 i \\ 2 & -1.90291-1.81163 i \\ 3 & -3.23300-3.00433 i \\ 4 & -4.06884-3.94940 i \\ 5 & -5.10001-4.96837 i \\ 6 & -6.05119-5.85692 i \\ 7 & -7.09451-6.88121 i \\ 8 & -8.03695-7.91822 i \\ 9 & -9.07841-8.95282 i \\ 10 & -10.0648-9.96230 i \\ 11 & -11.0449-10.9348 i \\ 12 & -12.0518-11.9671 i \\ 13 & -13.0490-12.9660 i \\ 14 & -14.0285-13.9565 i \\ 15 & -15.0394-14.9589 i \end{array} \right)$$