  Several quantities related to the Zernike circle polynomials admit an
expression as an infinite integral involving the product of two or three Bessel
functions. In this paper these integrals are identified and evaluated
explicitly for the cases of (a) the expansion coefficients of
scaled-and-shifted circle polynomials, (b) the expansion coefficients of the
correlation of two circle polynomials, (c) the Fourier coefficients occurring
in the cosine representation of the circle polynomials, (d) the transient
response of a baffled-piston acoustical radiator due to a non-uniform velocity
profile on the piston.
