Legendre polynomial spherical harmonics

A few examples of the application of the 3-1 symbols to calculations with real spherical harmonics will be shown. Frequently surface harmonics occur that are normalized like Legendre polynomials such harmonics will be denoted by the letter ( ,... 

The spherical harmonics are defined in terms of the associated Legendre polynomials, of variable cos 6, and exponential functions in... 

Relationship of spherical harmonics to associated Legendre polynomials... 

To motivate our next step recall that the LOFF spectrum can be view as a dipole [oc Pi (a )] perturbation of the spherically symmetrical BCS spectrum, where Pi(x) are the Legendre polynomials, and x is the cosine of the angle between the particle momentum and the total momentum of the Cooper pair. The l = 1 term in the expansion about the spherically symmetric form of Fermi surface corresponds to a translation of the whole system, therefore it preserves the spherical shapes of the Fermi surfaces. We now relax the assumption that the Fermi surfaces are spherical and describe their deformations by expanding the spectrum in spherical harmonics [17, 18]... 

The projection of T,p on each of the radial unit vectors can be evaluated in terms of the basic angular functions which make up the vector spherical harmonics.(27) Although these functions are associated Legendre polynomials for an arbitrarily oriented donor dipole, for the case of full azimuthal symmetry shown in Figure 8.19 the angular functions are ordinary Legendre functions, P (i.e., w = 0). Under these circumstances,... 

Since the associated Legendre polynomials (and the spherical harmonics) form an orthogonal set, only terms with / = / and m = m do not vanish in the integral of Eq. (3.41). Furthermore, for the 6 integration,... 

For clarity, in this chapter, we discuss the m=0 states in detail. The m 0 states are discussed in an example, although the formalism is equally applicable to all /, m. For m = 0 states, the spherical harmonics Yi reduce to Legendre polynomials. Pi ( cos 6). The first three of them are ... 

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. 

The 4>) function turns out to be an exponential and the ( ) function consists of Legendre polynomials. Their product () ( ) gives the spherical harmonic functions which Arfken writes as Y 6, ). Then, from Eq. 20.56,... 

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 three-particle distribution function g3(r,s) can be expressed in a series of Legendre polynomials [63]. Then expressing the Legendre polynomials in terms of spherical harmonics, we can write the expression for g3(r,s) as... 

To evaluate the averages like those in Eq. (4.78), it is very convenient to pass from cosines ((en)k) to the set of corresponding Legendre polynomials for which a spherical harmonics expansion (addition theorem)... 

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

We now expand the Legendre polynomial (cos 0ip) using the spherical harmonic... 

The functions Qim(9) and consequently the spherical harmonics Yim(6, associated Legendre polynomials, whose definition and properties are presented in Appendix E. To show this relationship, we make the substitution of equation (5.42) for cos 6 in equation (5.51) and obtain... 

The functions lVm(r) are the spherical harmonics, which are given in terms of the associated Legendre polynomials P m(x). For m > 0... 

All the correlation functions above are normalized, therefore equations (4 and 5) are identical to correlation functions over linear momentum p = mv and angular momentum J — lu, respectively. Note that, in this context I is the moment of inertia tensor The correlation function in equation (6) is calculated over the spherical harmonics. If m = 0, this reduces to time correlation function over Legendre polynomials ... 

Applying the addition theorem, the Legendre polynomials P/(cos0p) can be expressed in terms of products of the spherical harmonics as below... 

In such cases, Y is a spherical harmonic and w is a product of a power of r, an exponential function, aud a Legendre polynomial in r. We want to take a snapshot so time is fixed and we only care about the spatial coordinates. If we want to visualize these solutions, it is useful to think about where the functions are zero and what sorts of symmetry they have. 

The shim coils are constructed based on an expansion of the magnetic field inhomogeneity in terms of spherical harmonics. The fields produced by the coils are orthogonal and can be adjusted independent of each other. Following (2.1.18) the shim fields Bs m are characterized by the associated Legendre polynomials P m (cos 6) and the expansion... 

After the reconstruction, a cross-check should show that the reference reflection is degenerated to a 5-distribution, and there are no negative intensities in the desmeared image. If this is not the case, the found reference peak was broadened not only by imperfect orientation . In this case an iterative trial-and-error method is helpful the peak is proportionally narrowed, until over-desmearing can no longer be detected. The equations mentioned are directly applicable in case of fiber symmetry. If the symmetry of the scattering pattern is lower, the simplification must be reverted to a set of equations in the complete spherical harmonics instead of the Legendre polynomials. 

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