Legendre polynomials table

The functions are the associated Legendre polynomials of which a few are given in Table 1.1. They are independent of Z, the nuclear charge number, and therefore are the same for all one-electron atoms. 

For each choice of n (the number of points), the w, and the n zeros ( ) of the nth degree Legendre polynomial must be determined by requiring that the approximation be exact for polynomials of degree less than 2n + 1. These have been determined for n = 2 through 95 and an abbreviated table for some n is given in Table 1-16. The interval - 1 < < I is transformed onto the interval a < X < b by calculating for each x, (k = 1,. . ., n)... 

The variation of Pn(ft) with ft for a few values of n is sliown in Fig. 5. Since, in most physical problems, the Legendre polynomial involved is usually / tl(cos 0) we have shown in Fig. fi the variation of this function with 0. Numerical values may be obtained from Tables of Associated Legendre Functions (Columbia University Press, 194-5),... 

In Table 2.2 we have listed the first few spherical harmonics, for the s, p, and d states. It is worth noting that some authors introduce a factor of [—l]m in defining the associated Legendre polynomials, producing a corresponding difference in the spherical harmonics.2 There are i-m nodes in the 6 coordinate, and none in the <(> coordinate. 

The im (9) functions are related to the associated Legendre polynomials, and the first few are listed in table 6.1. The R i(r) are the radial wave functions, known as associated Laguerre functions, the first few of which are listed in table 6.2. The quantities n, l and m in (6.8) are known as quantum numbers, and have the following allowed values ... 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

In this expression, P2 is the second Legendre polynomial and i(t) is a unit vector with the same orientation as the transition dipole at time t. The brackets indicate an ensemble average over all transition dipoles in the sample. The correlation function has a value of one at very short times when the orientation of y(t) has not changed from its initial orientation. At long times, the correlation function decays to zero because all memory of the initial orientation is lost. At intermediate times, the shape of the correlation function provides detailed information about the types of motions taking place. Table I shows the three theoretical models for the correlation function which we have compared with our experimental results. 

Tables 9.5 and 9.6 give some approximations for (o Xo ) and (O )(60) along with the applicable constraints. The wavelength X is that for the wave traveling in the fluid, and if nf 1, then X = X /nf, where X0 is for travel in vacuum. Table 9.6 shows the various approximations used to represent the phase function (O )(6o) in terms of Legendre polynomials.

TABLE 9.6 Phase Function in Terms of Legendre Polynomials... 

It should be observed that in contrast to the Hermite and Legendre polynomials, the Laguerre polynomials contain both odd and even powers of x. The first few are given in Table 22.1. 

Table 8.6 Roots ( Kj) for Shifted Legendre Polynomials (Source Finlayson, 1972)...

The 0 part of the differential in equation 11.46 does have a known solution. The solution is a set of functions known as associated Legendre polynomials. (As with the Hermite polynomials, differential equations of the form in equation 11.46 had been previously studied, by the French mathematician Adrien Legendre, but for different reasons.) These polynomials, listed in Table 11.3, are functions of 0 only, but have two indices labeling the functions. One of the indices, an integer denoted , indicates the maximum power, or order, of 0 terms. (It also indicates the total order of the combination of cos 0 and sin 0 terms.) The second index, m, specifies which... 

Show that the wavefunction i is normalized over all space. Use the associated Legendre polynomial listed in Table 11.3. 

This is the associated Legendre polynomial from Table 11.3 combined with the g+im(i> jjjg proper value of m. 

Values for the associated Legendre polynomials and the normalized polar wavefunction 0j are listed in Table 2.1. 

The explicit expressions for the normalized associated Legendre polynomials are given in Table 41 for Z = 0, 1, 2, 3. The normalized spherical harmonics [Pg.58]

Table 4.5 Roots of Legendre polynomials and the weight factors for the Gauss-Legendre quadrature...

When Pi x) is chosen to be the Legendre set of orthogonal polynomials [see Table 3.7], the weight w(a ) is unity. The standard interval of integration for Legendre poiynomials is [-1, 1]. The transformation equation (4.92) is used to transform the Legendre polynomials to the interval [ji , which applies to our problem at hand ... 

The normalized associated Legendre polynomials, 0(0) are given in the following expression, and the first several are listed in Table 3-1. 

Table 3-1. The normalized associated Legendre polynomials, 0jj ( )and spherical harmonic wavefunctions, up to/=3...

In Table 6.2, the associated Legendre polynomials are listed for / < 3. We note that, for odd integers m, the associated Legendre polynomials are not true polynomials in x. Rather, they are generalizations of the Legendre polynomials... 

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