## 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