= Exponential moment of a Gaussian linear functional
If a linear functional $Z=\int a(\mathbf r)V(\mathbf r)\,d\mathbf r$ of a real <Gaussian random field> exists in mean square, it is a <Gaussian random variable>. Completing the square in its density gives
$$
\mathbb E e^Z=\exp\left(\mathbb EZ+\frac12\operatorname{Var}Z\right),\qquad \operatorname{Var}Z=\iint a(\mathbf r)a(\mathbf r')\operatorname{Cov}[V(\mathbf r),V(\mathbf r')]\,d\mathbf r\,d\mathbf r'.
$$
For a complex linear functional $\phi$, apply this to $Z=2\operatorname{Re}\phi$; then $\tfrac12\operatorname{Var}Z=\mathbb E|\phi-\mathbb E\phi|^2+\operatorname{Re}\mathbb E(\phi-\mathbb E\phi)^2$.
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