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

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

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 assymptotic form, but I'm not convinced...

Thanks in advance :-)

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...

Thanks in advance :-)