Complex Gaussian exponential moment ($\mathbb{E}[e^Z]=\exp (\mathbb{E}[Z^2]/2)$)

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.