Legendre polynomial equation

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. 

The goal of this appendix is to prove that the restrictions of harmonic polynomials of degree f to the sphere do in fact correspond to the spherical harmonics of degree f. Recall that in Section 1.6 we used solutions to the Legendre equation (Equation 1.11) to dehne the spherical harmonics. In this appendix we construct bona hde solutions to the Legendre equation then we show that each of the span of the spherical harmonics of degree is precisely the set of restrictions of harmonic polynomials of degree f to the sphere. 

Proof. First we will show that the Legendre polynomial of degree satisfies the Legendre equation with m = 0. Then we will deduce that for any m =... 

A solution F = R(r) (single valued in (p only if is a Legendre s polynomial of the nth degree in which case R satisfies the equation,... 

In the isotropic phase, differential Equation 2.106 is simplified to the Legendre equation with eigenfunctions, the Legendre polynomials Pn(cos9) and the eigenvalues n(n +1). 

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. 

In order to make our shortcut with m = 0 pay off, we need to show that the associated Legendre polynomials satisfy the general Legendre equation. Consider P x) = 3(1 — y ). 

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. 

According to Roe s analyses, two kinds of orientation distribution functions given by equations (1) and (2) may be expanded into infinite series of normalized and generalized spherical harmonics as equations (4) and (5). Amnii) and Tlf(Cj) in equations (4) and (5) are the normalized associated Legendre s polynomials as given by equations (6) and (7). and in equations... 

Equation (E.13) relates the associated Legendre polynomial Pfip) to the (/ -I- w)th-order derivative in equation (5.58)... 

The first few Legendre polynomials may be readily obtained from equation (E.2) and are... 

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. 

In particular the even chain terminates when / is even, while the odd chain terminates when l is odd. The corresponding Frobenius solution then simplifies to a polynomial. It is called the Legendre polynomial of degree /, or Pi(x). The modified form of equation (16) becomes... 

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... 

For m an integer the P/"(x) are polynomials. The polynomials P,°(x) are identical with the Legendre polynomials. The first few associated Legendre polynomials for x = cos 9 (a common form of Legendre s equation) are ... 

Recurrence Relations for the Legendre Polynomials. If wc differentiate both sides of equation (13.2) with respect to k we Slave... 

Series of Legendre Polynomials. In certain problems of potential theory it is desirable to be able to express a given function in the form of a series of Legendre polynomials. Wc can readily show that this is possible in the case in which the given function is a simple polynomial. For example, from the equations (13.4a) we have... 

Using these equations, we can write one term in the expansion of the wavepacket in terms of the associated Legendre polynomials [Eq. (4.60)] as... 

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