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 has sample paths that are -1 times differentiable.
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:
- ^ Minasny, B.; McBratney, A. B. (2005). "The Matérn function as a general model for soil variograms". Geoderma. 128 (3–4): 192–207. doi:10.1016/j.geoderma.2005.04.003.
- ^ Rasmussen, Carl Edward (2006) Gaussian Processes for Machine Learning
- ^ Santner, T. J., Williams, B. J., & Notz, W. I. (2013). The design and analysis of computer experiments. Springer Science & Business Media.
- ^ Abramowitz and Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. ISBN 0-486-61272-4.