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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Integral of a Complex Gaussian