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The bilateral Laplace transform of a Gaussian function could be established as: $$e^{x^2/2}=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}e^{-xy}e^{-y^2/2} dy$$

Then what should be a similar relation for a Gaussian function with complex variable:$e^{z^2/2}$?

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    The usual Laplace transform uses $\int_{0}^\infty$. Here you have a bilateral Laplace transform.2012-08-17
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    I wrote what I think is an elegant account of how to find this (bilateral) Laplace transform, but that's not necessarily on topic for your final question, so I deleted it. If you're interested, I'll un-delete it.2012-08-17
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    @RobertIsrael : If I were to call it a "two-sided Laplace transform", would that be too Germanic for you?2012-08-18
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    I have nothing against "two-sided". Too Germanic would be "zweiseitig".2012-08-19

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