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2CMU 9-tuple focus absolute values absolutely convergent Analytic Coefficients analytic solutions analytic throughout approximation BOCHER Cartesian Oval compute connected region continuous function continuous real functions convergent series convergent throughout corresponding terms counts curve derivative vanishes equa Equations with Analytic Existence Theorem foci invert focus at infinity follows at once four tangents fourth focus inequality infinite initial conditions isfy last section LINEAR DIFFERENTIAL EQUATIONS nodal bicircular quartic Non-Singular Points positive constant positive quantity proof given proof just given prove the theorem q are analytic q are continuous q are real quantity satisfying real analytic functions real foci real single finite real triple focus real variable region throughout satisfies equation satisfies the differential satisfies the initial series converges single finite foci single foci singular point solutions obtained take the limit THEORY OF LINEAR throughout an interval throughout the interval tion vanish throughout whole interval wish to prove