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Spectral Audio Signal Processing
    Gaussian Function Properties
       Gaussian Integral with Complex Offset

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Gaussian Integral with Complex Offset



Theorem:

$\displaystyle \int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}}, \quad p,c\in {\bf C},\;$re$\displaystyle \left\{p\right\}>0 $



Proof: When $ c=0$, we have the previously proved case. For arbitrary $ c=a+jb\in{\bf C}$ and real number $ T>\left\vert a\right\vert$, let $ \Gamma_c(T)$ denote the closed rectangular contour $ z = (-T) \to T \to (T+jb) \to (-T+jb) \to (-T)$, depicted in Fig.D.1. figure[htbp] \includegraphics{eps/gammarect} Clearly, $ f(z) \isdef e^{-pz^2}$ is analytic inside the region bounded by $ \Gamma_c(T)$. By Cauchy's theorem [41], the line integral of $ f(z)$ along $ \Gamma_c(T)$ is zero, i.e.,

$\displaystyle \oint_{\Gamma_c(T)} f(z) dz = 0 $

This line integral breaks into the following four pieces:

\begin{eqnarray*} \oint_{\Gamma_c(T)} f(z) dz &=& \underbrace{\int_{-T}^T f(x) ... ...{-T} f(x+jb) dx}_3 + \underbrace{\int_{b}^{0} f(-T+jy) jdy}_4\\ \end{eqnarray*}

where $ x$ and $ y$ are real variables. In the limit as $ T\to\infty$, the first piece approaches $ \sqrt{\pi/p}$, as previously proved. Pieces $ 2$ and $ 4$ contribute zero in the limit, since $ e^{-p(t+c)^2} \to 0$ as $ \left\vert t\right\vert\to\infty$. 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 $ T\to\infty$,

$\displaystyle \int_{-\infty}^\infty f(x+jb) dx = \sqrt{\frac{\pi}{p}}. $

Making the change of variable $ x=t+a=t+c-jb$, we obtain

$\displaystyle \int_{-\infty}^\infty f(t+c) dt = \sqrt{\frac{\pi}{p}} $

as desired.

Previous: Area Under a Real Gaussian
Next: Fourier Transform of Complex Gaussian

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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