This paper presents exact analytic formulae for the eclipse of a star with quadratic or nonlinear limb darkening. These formulae are used to compute lightcurves for planetary transits, which are essential for detecting and characterizing exoplanets. The results are applied to the Hubble Space Telescope (HST) observations of HD 209458, yielding a planetary radius-to-stellar radius ratio of 0.1207 ± 0.0003. The formulae allow for fast and accurate computation of lightcurves using limb-darkening coefficients from model atmospheres, aiding in the detection, simulation, and parameter fitting of planetary transits. The paper discusses the effects of limb-darkening on the lightcurve of a star during an eclipse. Limb-darkening causes the star to be more centrally bright, leading to more significant dimming during an eclipse and creating curvature in the trough. The nonlinear limb-darkening law, which fits well a wide range of stellar models and observational bands, is presented. The paper derives analytic functions for transit lightcurves under both quadratic and nonlinear limb-darkening laws, and provides a simpler form for the quadratic case. For small planets (p ≲ 0.1), an approximation is given that is very fast to compute and fairly accurate. The paper also discusses the application of these formulae to the HST observations of HD 209458, where the best-fit parameters are determined. The results show that the quadratic law is sufficient for main-sequence stars when an accuracy of less than 3% is required. The paper concludes that the derived analytic expressions for eclipses with quadratic and nonlinear limb-darkening are accurate and efficient, and that they can be used for simulating planetary transit searches and for fitting and measuring the errors of detected planetary transit events. The formulae presented can help distinguish true planetary transits from contaminants such as grazing eclipsing binaries and triple systems. The code developed in this paper is available for download and can be used for planetary transit searches.This paper presents exact analytic formulae for the eclipse of a star with quadratic or nonlinear limb darkening. These formulae are used to compute lightcurves for planetary transits, which are essential for detecting and characterizing exoplanets. The results are applied to the Hubble Space Telescope (HST) observations of HD 209458, yielding a planetary radius-to-stellar radius ratio of 0.1207 ± 0.0003. The formulae allow for fast and accurate computation of lightcurves using limb-darkening coefficients from model atmospheres, aiding in the detection, simulation, and parameter fitting of planetary transits. The paper discusses the effects of limb-darkening on the lightcurve of a star during an eclipse. Limb-darkening causes the star to be more centrally bright, leading to more significant dimming during an eclipse and creating curvature in the trough. The nonlinear limb-darkening law, which fits well a wide range of stellar models and observational bands, is presented. The paper derives analytic functions for transit lightcurves under both quadratic and nonlinear limb-darkening laws, and provides a simpler form for the quadratic case. For small planets (p ≲ 0.1), an approximation is given that is very fast to compute and fairly accurate. The paper also discusses the application of these formulae to the HST observations of HD 209458, where the best-fit parameters are determined. The results show that the quadratic law is sufficient for main-sequence stars when an accuracy of less than 3% is required. The paper concludes that the derived analytic expressions for eclipses with quadratic and nonlinear limb-darkening are accurate and efficient, and that they can be used for simulating planetary transit searches and for fitting and measuring the errors of detected planetary transit events. The formulae presented can help distinguish true planetary transits from contaminants such as grazing eclipsing binaries and triple systems. The code developed in this paper is available for download and can be used for planetary transit searches.