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Subsections
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.
We first consider the simplelooking indefinite integral

(128) 
which, surprisingly at first glance, has no solution among the
elementary functions. However, we can find the definite integral

(129) 
which is called a onedimensional Gaussian integral by squaring it and
converting it to polar coordinates to obtain

(130) 
The general dimensional Gaussian integral can be written as

(131) 
where is a dimensional vector and is a
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 2dimensional case for the purpose of this
thesis.
We now consider the twodimensional Gaussian integral

(132) 
(so that the vector
and
)
which can be done by just completing the square and using the above
formula. The result is

(133) 
In the last form, the answer is valid for any number of dimensions
( is the number of dimensions).
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Marakani Srikant
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