Gaussian Integral with Complex Offset

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The complex plane may be defined as the graph of all complex numbers extit{z} = extit{x} + extit{j y} formed by using the real part extit{x} as the horizontal coordinate and the imaginary part extit{y} as the vertical coordinate on the graph. In other words, each complex number extit{z} = extit{x} + extit{j y} corresponds to a point in the complex plane located at Cartesian coordinate ( extit{x,y}). — Click for https://ccrma.stanford.edu/~jos/mdft/Complex_Plane.html

The set of real numbers may be expressed informally as the set of all numbers that may be expressed as a decimal number with possibly an infinite number of digits. More formally, the set of real numbers is the field of rational and irrational numbers. — Click for http://mathworld.wolfram.com/RealNumber.html

The Gaussian probability density function is, in Matlab syntax, exp(-(x-mu)^2/(2*sigma^2))/sqrt(2*pi*sigma^2), where mu is the mean and sigma is the standard deviation. — Click for http://en.wikipedia.org/wiki/Normal_distribution

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Spectral Audio Signal Processing

Gaussian Function Properties

Area Under a Real Gaussian

Fourier Transform of Complex Gaussian

Gaussian Integral with Complex Offset

Theorem:

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

(D.12)

Proof: When , we have the previously proved case. For arbitrary $c=a+jb\in\mathbb{C}$ 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 D.1:** Contour of integration in the complex plane.
$\includegraphics[width=0.8\twidth]{eps/gammarect}$

Clearly, $f(z) \isdef e^{-pz^2}$ is analytic inside the region bounded by $\Gamma_c(T)$ . By Cauchy's theorem [42], the line integral of along $\Gamma_c(T)$ is zero, i.e.,

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

(D.13)

This line integral breaks into the following four pieces:

$\begin{eqnarray*} \oint_{\Gamma_c(T)} f(z) dz &=& \underbrace{\int_{-T}^T f(x) dx}_1 + \underbrace{\int_{0}^{b} f(T+jy) jdy}_2\\ &+& \underbrace{\int_{T}^{-T} f(x+jb) dx}_3 + \underbrace{\int_{b}^{0} f(-T+jy) jdy}_4\\ \end{eqnarray*}$

where and are real variables. In the limit as $T\to\infty$ , the first piece approaches $\sqrt{\pi/p}$ , as previously proved. Pieces and 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 $T\to\infty$ ,