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Subsections

Gaussian Integrals

For the reader who might be unacquainted with Gaussian integrals, we present some of the main results about them. Obviously, this is in no way an exhaustive or thorough presentation on the subject but should suffice for the understanding of the chapter.

One-Dimensional Gaussian Integrals

We first consider the simple-looking indefinite integral
\begin{displaymath}
\int e^{-ax^2}dx
\end{displaymath} (128)

which, surprisingly at first glance, has no solution among the elementary functions. However, we can find the definite integral
\begin{displaymath}
\int_{-\infty}^\infty e^{-ax^2}dx
\end{displaymath} (129)

which is called a one-dimensional Gaussian integral by squaring it and converting it to polar coordinates to obtain
\begin{displaymath}
\int_{-\infty}^\infty e^{-ax^2}dx = \sqrt{\int_{-\infty}^\in...
...ty
\int_0^{2\pi} re^{-ar^2}dr d\theta} = \sqrt{\frac{\pi}{a}}
\end{displaymath} (130)

Higher-Dimensional Gaussian Integrals

The general $n$-dimensional Gaussian integral can be written as
\begin{displaymath}
\int e^{-\tilde{\v{x}}A\v{x}} d^n\v{x}
\end{displaymath} (131)

where $\v{x}$ is a $n$-dimensional vector and $A$ is a $n \times n$ matrix which we can make symmetric since multiplication is a commutative operation. While a solution for the integral exists, we only need to consider the 2-dimensional case for the purpose of this thesis.

We now consider the two-dimensional Gaussian integral

\begin{displaymath}
\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(ax^2+2bxy+cy^2)} dx
dy
\end{displaymath} (132)

(so that the vector $\v{x} = \left( \begin{matrix}x\\ y\\ \end{matrix}\right)$ and $A = \left(
\begin{matrix}
a & b\\
b & c\\
\end{matrix} \right)$ ) which can be done by just completing the square and using the above formula. The result is
\begin{displaymath}
\begin{split}
&\int_{-\infty}^\infty \int_{-\infty}^\infty e...
...{\sqrt{ac-b^2}} = \frac{\pi^{m/2}}{\sqrt{\det(A)}}
\end{split}\end{displaymath} (133)

In the last form, the answer is valid for any number of dimensions ($m$ is the number of dimensions).


next up previous contents
Next: The General Program Up: thesis Previous: Conclusion   Contents
Marakani Srikant 2000-08-15