The apodization problem is solved analytically for determining the pupil function (amplitude distribution over the exit pupil) which minimizes the second moment of the intensity distribution in the Fraunhofer diffraction pattern of a point object, on the conditions that the central intensity is prespecified to take a certain value less than that for the standard Airy-type optical system without filter and that the optical system is a passive one. The problem is treated for a rotationally symmetric optical system without aberrations. The method used to solve the problem is the calculus of variations of unconventional type involving the constraint of inequalities due to the passivity of the optical system. A detailed derivation of the optimum pupil function is presented and discussion on the resultant pupil function is given.