Weierstrass transform in complex variable

The usual Weierstrass transform of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as: $$e^{D^2/2}f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-yD}f(x)e^{-y^2/2} dy=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x-y)e^{-y^2/2}dy$$ where $D=\frac{d}{dx}$.

Now if $D$ is with respect to complex variable $z$, how will the Weierstrass transform be different from the one above?

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