# Solving complicated equation involving integral of error functions

I've been trying to solve the following equation for $\sigma$

$$\sigma^2 = \int_0^1 \left\{ \frac{1}{2} \operatorname{erfc} \left[ \frac{x + B}{\sqrt{2 (Rp - 1)} \, \sigma} \right] + \frac{1}{2} \operatorname{erfc} \left[ \frac{x - B}{\sqrt{2 (Rp - 1)} \, \sigma} \right] \right\}^{p - 1} \, x \, dx$$

with $B$, $p$ and $R$ fixed. In particular, I'm interested in the $B \to 0$ and $p \to \infty$ limits, but I'd like to take these limits only after solving the equation.

Any insight on how to do it? I've tried solving it numerically for some specific cases and seem to have found an asymptotic form, but I'm not convinced...

• You need some more restrictions for this to really make sense. For example $Rp \not = 1$. – Benjamin Apr 24 '15 at 19:24
Not a full answer (if such is possible at all), but note that $\text{erfc}(\cdot)$ is $O(1)$, so your expression becomes $$\sigma^2 = c^{p -1}\frac{1}{2}$$ and so $\sigma^2$ either blows up to infinity or converges to 0.