## 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:

*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