## Gaussian Integral with Complex Offset

**Theorem: **

(D.12) |

*Proof: *
When
, we have the previously proved case. For arbitrary
and real number
, let
denote the closed rectangular contour
, depicted in Fig.D.1.

Clearly,
is analytic inside the region bounded
by
. By Cauchy's theorem [42],
the line integral of
along
is zero, *i.e.*,

(D.13) |

This line integral breaks into the following four pieces:

where
and
are real variables. In the limit as
,
the first piece approaches
, as previously proved.
Pieces
and
contribute zero in the limit, since
as
. Since the total contour integral is
zero by Cauchy's theorem, we conclude that piece 3 is the negative of
piece 1, *i.e.*, in the limit as
,

(D.14) |

Making the change of variable , we obtain

(D.15) |

as desired.

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Fourier Transform of Complex Gaussian

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Integral of a Complex Gaussian