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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Gaussian Integral with Complex Offset
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Infinite Flatness at Infinity