This tag is for questions relating to Legendre Functions (or Legendre Polynomials), solutions of Legendre's differential equation (generalized or not) with non-integer parameters.
The second order differential equation given as $$(1 − x^2)y'' − 2xy' + α(α + 1)y = 0\qquad \text{with}~~~~n > 0~, ~~~|x| < 1$$ is known as Legendre’s equation. The solutions of this equation are called Legendre Functions of degree $~n~$.
The Legendre Functions are often referred to as Legendre Polynomials $~P_n(x)~$.
Since Legendre's differential equation is a second order ordinary differential equation, two sets of functions are needed to form the general solution. Legendre Polynomials/Functions of the second kind $~Q_n(x)~$ are then introduced.
Legendre Polynomials/Functions of the first kind $$P_n(x)=\dfrac 1{2^n~n!}~\dfrac{d^n}{dx^n}(x^2-1)^n~.$$ Legendre Polynomials/Functions of the second kind $$Q_n(x)=\dfrac 1{2}~P_n(x)~\ln\dfrac{1+x}{1-x}~.$$
Applications: Legendre functions are important in mathematics as well as physics. Legendre polynomials show up in spherical harmonics, which are eigen function of the Laplace operator over the sphere. This decomposition is actually used in meteorological spectral models, for example the Global Forecast System and Integrated Forecast System. Legendre functions are also widely used in determination of wave functions of electrons in the orbits of an atom and in the determination of potential functions in the spherical symmetric geometry etc. Also nuclear reactor physics, Legendre polynomials have an extraordinary importance.
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