Multi-dimensional Gaussian Function
In an -dimensional space a Gaussian function can be defined as
where is a column of coordinates, is a positive-definite matrix, and denotes transposition.
The integral of a Gaussian function over the whole -dimensional space is given as
It can be easily calculated by diagonalizing the matrix and changing the integration variables to the eigenvectors of .
More generally a shifted Gaussian function is defined as
where is the shift vector and the matrix can be assumed to be symmetric, . The following integrals with this function can be calculated with the same technique,
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