I am trying to solve this equation for $\mu$ $$ (2a-1) \mu \cdot \mathrm{erf}\left( \frac{\mu}{\sqrt{2} \sigma} \right) + \frac{2\sigma \cdot (2a-1) \cdot e^{-\frac{\mu^2}{2 \sigma^2}} }{\sqrt{2\pi}} + \mu=0 $$ with $0 \le a \le 1$ and $\sigma > 0$. Clearly, when $a=\frac{1}{2}$ the first two terms cancel out and the solution is $\mu=0$.
I can find the solution numerically, and I think it should be possible to express the solution as a linear function of $\sigma$, something like $\mu=f(a)\cdot \sigma$, with $f(0.5)=0$, but I don't know how to proceed.
Any help is appreciated!