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So if we have a two dimensional Gaussian function $\frac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}$, then the following integration $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}dxdy=1$$

Now if we switch to complex variables $z,z^*$, what should be the range of those complex variables for the same integration above transformed into complex variable?

$$\int\int\frac{i}{2}\frac{1}{2\pi}e^{-\frac{|z|^2}{2}}dzdz^*=1$$

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    You would have just one complex variable, and you'd integrate over the entire complex plane. I think.2012-08-27
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    The measure on the complex plane is not a product measure, and the integral of the probability density is not an iterated integral. It becomes one when $\mathbb{C}$ is mapped into $\mathbb{R}^2$.2012-08-27
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    So how to evaluate this Gaussian function in complex plane and get the same result?2012-08-27
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    I'm not sure how this goes, but I think it is exactly the same since the function $$e^{-|z|^2/2}$$is real...2012-08-27
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    One evaluates the integral by mapping to $\mathbb{R}^2$.2012-08-27

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