## 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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