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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 :-)

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  • $\begingroup$ You need some more restrictions for this to really make sense. For example $Rp \not = 1$. $\endgroup$ – Benjamin Apr 24 '15 at 19:24
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    $\begingroup$ It seems extremely implausible that in general there would be a closed form for the integral, much less a closed-form solution to the equation, if that's what you're asking. $\endgroup$ – Robert Israel Apr 24 '15 at 20:22
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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.

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