# Formulas for the multivariate Gaussian function?

I wonder if anyone has formulas for the derivatives of $f(x)=(2\pi)^{-n/2}|\Sigma|^{-1/2}\exp \{-\frac{1}{2}x^t \Sigma^{-1} x\}$ or for instance, for,

$$\int_{\mathbb{R}^n}D^{\alpha}f(x)dx$$

is there any website or book where I can find these?

Thanks you very much!

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If you convert the density function to use the actual elements of $x$ and $\Sigma$ rather than the matrix equivalents, taking derivatives will become more clear. –  Max Oct 26 '12 at 16:39
Will this involve Hermite polynomials? –  Michael Hardy Oct 26 '12 at 17:34

## 1 Answer

The 2D ones are used all the time in machine vision: e.g. http://bmia.bmt.tue.nl/education/courses/fev/book/pdf/04%20Gaussian%20derivatives.pdf

The paper says:

The two-dimensional Gaussian derivative function can be constructed as the product of two one-dimensional Gaussian derivative functions, and so for higher dimensions, due to the separability of the Gaussian kernel for higher dimensions.

So basically, just the do the 1D case and scale appropriately.

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The 1D case being given by Hermite polynomials times the Gaussian. –  Lucas Dec 11 '12 at 7:06