Legendre functions

Examples 10.1 to 10.20

Published in L. M. B. C. Campos, Classification and Examples of Differential Equations and their Applications, 2019

*standard CDXXXII. Spherical harmonics as the solution of the associated Legendre differential equation (10.595c) finite in the interval |x|≤1, which is proportional to the first associated Legendre function of the first kind (10.595a), with integer degree ν=n larger n>m than the integer order μ=m (note 9.22);

Estimating the expansion coefficients of a geomagnetic field model using first-order derivatives of associated Legendre functions

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Journal Information

Published in Optimization Methods and Software, 2018

H. Martin Bücker, Johannes Willkomm

The associated Legendre functions are commonly used in various areas of mathematics and physical sciences. This has led to several normalization schemes that are adapted to suit the special purposes of the different scientific communities. Here, the focus is on three different normalization schemes. The functions given in the scheme (1) are called unnormalized. A second normalization scheme consists of Schmidt seminormalized associated Legendre functions defined by Thus, the Schmidt seminormalized associated Legendre functions coincide with the unnormalized functions for m=0. The third normalization scheme is given by where the functions are called normalized associated Legendre functions.

Legendre functions

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Examples 10.1 to 10.20

Estimating the expansion coefficients of a geomagnetic field model using first-order derivatives of associated Legendre functions