The expectation value of one side truncated (upper tail) normal distribution is defined as follows:
$$ \operatorname{E}(X \mid X>a) = \mu +\sigma\lambda(\alpha) \!$$ where $$ \alpha=(a-\mu)/\sigma,\; \lambda(\alpha)=\phi(\alpha)/[1-\Phi(\alpha)], $$
$$ \phi(\xi)=\frac{1}{\sqrt{2 \pi}}\exp{\left(-\frac{1}{2}\xi^2\right)}, $$ and $\Phi(\cdot)$ is the cumulative normal distribution function.
My question:
Given $\operatorname{E}(X \mid X>a)$, $\mu$, and $a$, how do I obtain $\sigma$ using an analytic formula (non-iterative)?