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:
![\begin{eqnarray*}
\oint_{\Gamma_c(T)} f(z) dz
&=& \underbrace{\int_{-T}^T f(x) dx}_1
+ \underbrace{\int_{0}^{b} f(T+jy) jdy}_2\\
&+& \underbrace{\int_{T}^{-T} f(x+jb) dx}_3
+ \underbrace{\int_{b}^{0} f(-T+jy) jdy}_4\\
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2765.png)
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
![$ x=t+a=t+c-jb$](http://www.dsprelated.com/josimages_new/sasp2/img2770.png)
![]() |
(D.15) |
as desired.
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Integral of a Complex Gaussian