= Solution
Apply the <isotropic cosmological correlation-power-spectrum relation> and put $K=k_{\max}$, $q=Kr$. The <band-limited linear density correlation> is
$$
\xi(r)=\frac{A}{2\pi^2r}\int_0^K k^2\sin(kr)\,dk.
$$
Two <integrations by parts> give the primitive $-k^2\cos(kr)/r+2k\sin(kr)/r^2+2\cos(kr)/r^3$. Evaluating both endpoints yields
$$
\boxed{\xi(r)=\frac{A}{2\pi^2r^4}
[-q^2\cos q+2q\sin q+2(\cos q-1)],\qquad r>0.}
$$
The apparent singularity at the origin is removable. Either the <Taylor series> or the original integral gives
$$
\boxed{\xi(0)=\frac{AK^4}{8\pi^2}.}
$$
In fact $\xi(r)=\xi(0)[1-q^2/9+q^4/240+O(q^6)]$. The sharp spectral cutoff produces oscillations and negative correlations at some nonzero separations, even though the <matter power spectrum> is everywhere nonnegative. A <covariance function> need not be pointwise nonnegative; its required positivity is that of the covariance quadratic form.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-41-cutoff-correlation.png]
{title=An abrupt cutoff in a nonnegative density power spectrum produces an oscillating correlation function}
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