Renormalization and its optimization of the Legendre function of the sec-ond kind
The series expansion of ellipsoidal harmonic functions is the basis for ellipsoid harmonic modeling of the Earth's gravity field.However,the main difficulty in dealing with ellipsoidal harmonics series lies in the calculation of Legendre func-tions of the second kind.Jekeli's renormalization method simplifies this calculation process.Based on Jekeli's renormalization,this paper deduces two optimization recursive methods based on transformations of Gaussian hypergeometric functions are de-rived in details.At the same time,these two optimization recursive methods are used to calculate the second type of Legendre function,and expand it to the second derivative.Numerical calculations have proven that the optimization recursive method can effectively accelerate convergence,shorten calculation time,and is applicable to higher orders,which makes the ellipsoid har-monic function series more convenient and feasible in practical applications.
the Legendre function of the second kindthe associated Legendre differential equationrenormalizationGaussian hypergeometric function
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