The Matérn covariance between two points separated by d distance units is given by 
where is the gamma function, is the modified Bessel function of the second kind, and ρ and ν are non-negative parameters of the covariance.
A Gaussian process with Matérn covariance is times differentiable in the mean-square sense.
The power spectrum of a process with Matérn covariance defined on is the (n-dimensional) Fourier transform of the Matérn covariance function (see Wiener–Khinchin theorem). Explicitly, this is given by
Simplification for specific values of νEdit
Simplification for ν half integerEdit
When , the Matérn covariance can be written as a product of an exponential and a polynomial of order :
- for :
- for :
- for :
The Gaussian case in the limit of infinite νEdit
As , the Matérn covariance converges to the squared exponential covariance function
Taylor series at zero and spectral momentsEdit
The behavior for can be obtained by the following Taylor series:
When defined, the following spectral moments can be derived from the Taylor series:
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