Complex Gaussian exponential moment
= Complex Gaussian exponential moment
{title2=$\mathbb E[e^Z]=\exp(\mathbb E[Z^2]/2)$}
For a centered complex Gaussian variable $Z$ whose real and imaginary parts are jointly <Gaussian>, the displayed formula follows by analytic continuation of the real joint <moment-generating function>. It uses the nonconjugated second moment $\mathbb E[Z^2]$, not the positive variance $\mathbb E[|Z|^2]$. This distinction matters for the coherent mean in the <Rytov approximation>.