Legendre equation functions

Let us notice that due to orthogonality of Legendre s polynomials many functions can be represented as a series, which is similar to Equation (1.162), and this fact is widely used in mathematical physics. Now, we will derive the differential equation, one of the solutions of which are Legendre s functions. 

Let us demonstrate that with help of Legendre s functions we can find a solution of Laplace s equation. As is well known, Laplace s equation has the following form in the spherical system of coordinates ... 

The second equation of this set is a known equation (Chapter 1) and its solutions are Legendre s functions P (t]) and Now we demonstrate that the first... 

This function is a solution of Laplace s equation regardless of the values of constants, and our goal is to find such of them that the potential satisfies the boundary condition on the surface of the given ellipsoid of rotation and at infinity. In order to solve this problem we have to discuss some features of Legendre s functions. First of all, as was shown in Chapter 1, the Legendre s function of the first kind P (t]) has everywhere finite values and varies within the range... 

The Legendre s function of the second kind g (> ) has completely different behavior in particular, it tends to infinity when = 1. In accordance with Equation (2.121), this happens at points of the z-axis. Since the potential has everywhere a finite value the function Qn(ri) cannot describe the attraction field and has to be removed from Equation (2.132). This first simplification gives ... 

Proposition A.l The Legendre functions of Definition A.1 satisfy the Legendre equation. 

Next we fix an integer m with 1 < m < and show that satisfies the Legendre equation (Equation A.l). Since the function P, o satisfies Equation A,2, we have... 

We do not discuss the associated Legendre equation, but give its solutions in Appendix F. The solutions are called associated Legendre functions and are derivatives of polynomials known as Legendre polynomials. 

Equation (F-46) is the same as the associated Legendre equation if K = lit + 1), where I is an integer that must be at least as large as m. The set of solutions is known as the associated Legendre functions, given for non-negative values of m by ... 

The purpose of this exercise is to show that the relatively simple solution of the m = 0 case of the Legendre equation can be extended to the general case for nonzero m values. The proof is given by Anderson [1] in two steps. First, define a function [f x) = (1 - x ) g(x)] and substitute it into the associated equation for nonzero values of m. 

This equation is known as the associated Legendre equation, and the function Uy denoted by u = Pn( )j is called the associated Legendre polynomial of degree n and order m. From equations 4 36 and 4 37, we see that... 

The Linear Differential Equation of the Second Order, 48. The Legendre Polynomials, 62. The Associated Legendre Polynomials, 52. The General Solution of the Associated Legendre Equation, 53. The Functions 0j.r ( ) and 57. Recursion Formulae for the Legendre Polynomials, 59. The Hermite Polynomials, 60. The Laguerre Polynomials, 63. 

Equation 54 implies that U is a function of S and P, a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Hehnholt2 energy,, and Gibbs energy, G ... 

Letting n = n n + 1) and choosing the minus sign at the right hand side of the second equality we obtain Legendre s equation one of its solutions is the function... 

The generating functions g " p, s) for the associated Legendre polynomials may be found from equation (E.l) by letting... 

At this point, we may proceed in one of two ways, which are mathematically equivalent. In the first procedure, we note that from the generating function (E.l) for Legendre polynomials Pi, equation (J.3) may be written as... 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

The solutions of differential equations often define series of related functions that can be obtained from simple generating functions or formulae. As an example consider the Legendre polynomials... 

To normalize a function such as P m(cos0) it is necessary to equate the integral / [P,m( cos 9)]2dx = 1. Starting from the Rodriques formula and integrating by parts it can be shown that the normalized associated Legendre functions are ... 

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