Continuous Limit of Multiple Gravitational Lens Effect and Average Magnification Factor

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We show that the gravitational magnification factor averaged over all configurations of lenses in a locally inhomogeneous universe satisfy a second order differential equation with redshift z by taking the continuous limit of multi-plane gravitational lens equation (the number N of lenses →∞) and that the gravitationally magnified Dyer-Roeder distance in a clumpy universe becomes to that of the Friedmann-Lematre universe for arbitrary values of the density parameter Ω₀ and of a mass fraction α (smoothness parameter).