## Integral of a Complex Gaussian

Theorem:

 (D.7)

Proof: Let denote the integral. Then

where we needed re to have as . Thus,

 (D.8)

as claimed.

### Area Under a Real Gaussian

Corollary: Setting in the previous theorem, where is real, we have

 (D.9)

Therefore, we may normalize the Gaussian to unit area by defining

 (D.10)

Since

 and (D.11)

it satisfies the requirements of a probability density function.

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