3
Legendre Polynomials and Legendre Functions
Abstract
Legendre polynomials and, respectively, Legendre functions are one of the most
important functions in physics. In this chapter, we will discuss and derive corresponding program codes supporting complex arguments and complex indices.
The code is freely available. The functions covered are Legendre polynomials
and Legendre functions of first and second kind, the evaluation of the nodes for
the Legendre functions of first kind based on the corresponding Jacobi matrix,
the Mehler or conical functions, the toroidal or ring functions, and others. An
application is, e.g., the evaluation of spherical harmonics. In dependence of the
function arguments and the function indices, the evaluation will be based either
on recurrence relations or on hypergeometric functions.
Topic of this chapter are Legendre polynomials and Legendre functions and related
functions in the real and complex domains. Legendre functions play an important
role in many areas in physics, e.g., in electrodynamics in solving the Laplace equation in spherical coordinates, in quantum dynamics with respect to spherical harmonics, Legendre functions with complex degrees in scattering theory, and so forth.
In the next chapter, we will list relevant functions, followed by chapters about
Legendre polynomials and Legendre functions. In Chap. 4, we will discuss complex
indices, the Mehler functions, and the toroidal functions.
3.1
Function Overview
Legendre polynomial:
• The MATLAB function P = legendre(n,x) computes the associate
Legendre functions of degree n and order m = 0 · · · n.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_3
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