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The Laplace integral method for a differential equation seeksDifferentiating under the integral and substituting intogivesSince $z{e}^{zt}={\partial }_t{e}^{zt}$, integration by parts yieldsIt is therefore enough to choose $f$ and $C$ so thatand the boundary term vanishes. Solving the first-order equation on $-1<t<1$ givesTake $C$ to be the interval $[-1,1]$. Although $f$ has integrable endpoint singularities,vanishes at both endpoints, so the boundary term is zero. ThusAt $z=0$, substituting $t=\sin \theta$ givesThe normalization $y(0)=1$ therefore sets $K=1/\pi$. By the defining regularity and normalization of the modified Bessel function, this proves the real integral representation of the modified Bessel function I0:

A zero-mean Gaussian process need not have zero average in each realization. For the periodic covariance function, the constant Fourier component iswhere $I_0$ is a modified Bessel function. Replacing $f(t)$ by $f(t)-P^{-1}{\int }_0^{P}f(u)du$ gives a Gaussian process with kernel $k_P(t,t^{\prime })-c_0$. This is a positive-semidefinite kernel because it is a covariance after a linear transformation, and each sample has zero period average. The real integral representation of the modified Bessel function I0 yields the displayed constant.