Harmonic polynomials

In (38), hi(uj) is an harmonic polynomial of order / in ui, U2 and M3, while hi sj) is the same harmonic polynomial with uj replaced by Sj. The Shibuya-Wulfman integrals can then be calculated by resolving the product of hyperspherical harmonics in (37) into terms of the form u hi(uj). To illustrate this second method for the evaluation of... 

Physicists are familiar with many special functions that arise over and over again in solutions to various problems. The analysis of problems with spherical symmetry in P often appeal to the spherical harmonic functions, often called simply spherical harmonics. Spherical harmonics are the restrictions of homogeneous harmonic polynomials of three variables to the sphere S. In this section we will give a typical physics-style introduction to spherical harmonics. Here we state, but do not prove, their relationship to homogeneous harmonic polynomials a formal statement and proof are given Proposition A. 2 of Appendix A. 

Relating the spherical harmonic functions introduced here to the homogeneous harmonic polynomials is not logically necessary in this book. Morally, however, the calculation is well worth doing, in the name of better communication between mathematics and physics. Because this calculation is a bit tricky, we have postponed it to Appendix A. 

A more interesting example involves the vector space P3 of complex-coefficient polynomials in three variables. Let H denote the subset of P3 containing only harmonic polynomials, i.e., only polynomials p in three variables satisfying = 0. Then H is a subspace of the vector space To... 

Restrictions of homogeneous harmonic polynomials play an important role in our analysis. 

Definition 2.6 Suppose f is a nonnegative integer. Define the vector space of homogeneous harmonic polynomials of degree f in three variables... 

In Section 7.1 we will use this characterization of homogeneous harmonic polynomials as a kernel of a linear transformation (along with the Fundamental Theorem of Linear Algebra, Proposition 2.5) to calculate the dimensions of the spaces of the spherical harmonics. 

All of the concepts of this section — kernel, image. Fundamental Theorem, homogeneous harmonic polynomials and isomorphisms — come up repeatedly in the rest of the text. 

Exercise 3.22 Show that the set of harmonic polynomials on is not closed under multiplication. (The point of this exercise is that in Chapter 7, when we wish to show that the restrictions of harmonic polyn omials to S " span S, we will not be able to appeal directly to the Stone-Weierstrass theorem. Rather, we will relate restrictions of harmonic functions to restrictions of polynomial functions and then appeal to the results of Section 3.5.)... 

Proof. Consider any function y e. By Definition 2,6, there is a homogeneous harmonic polynomial p of degree f such that y = PI52. Now rotating a polynomial preserves its degree (by Exercise 4.14), and the Laplacian is invariant imder rotation (by Exercise 3.11). So for any g e SO(3 ) the function g p is a homogeneous harmonic polynomial of degree . Hence g-y = g s2-p is an element of 3. ... 

The following consequence of Schur s lemma will be useful in the proof that every polynomial restricted to the two-sphere is equal to a harmonic polynomial restricted to the two-sphere (Proposition 7.3). The idea is that once we decompose a representation into a Cartesian sum of irreducibles, every irreducible subrepresentation appears as a term in the sum. 

In this proposition, as in Proposition 6.14, it is possible to drop the hypothesis that the representation be unitary. See Exercise 6.13. We will apply this classification of irreducibles of 50(3) in Section 7.1 to show that for each nonnegative integer n the set of homogeneous harmonic polynomials of degree n forms an irreducible representation of 50(3). 

Combining this last result with our knowledge of the classification of the irreducible representations of the group 50(3), we can show that the representation of the rotation group on homogeneous harmonic polynomials of any fixed degree is irreducible. 

Consider the polynomial (y — izY, a harmonic polynomial of degree f. Note that the polynomial is indeed harmonic, since all polynomials of degree zero or one are harmonic, while for I > 2,... 

Proposition 7.3 implies that any polynomial on the two-sphere in E can be written as a sum of harmonic polynomials. See Exercise 7.3. This fact is important to the proof of Proposition 7.4. The point is that we cannot apply the Stone-Weierstrass theorem directly to harmonic functions (see Exercise 3.22). However, we can apply the Stone-Weierstrass theorem to polynomials. Proposition 7.3 is the link we need. 

Recall the vector space y of spherical harmonics from Definition 2.6 y is the set of restrictions to of homogeneous harmonic polynomials. Recall also the definition of spanning (Defiiution 3.7). The set y of spherical harmoiucs spans... 

Exercise 7.4 Illustrate Proposition 7.3 hy finding a basis ofP consisting of five harmonic polynomials and one polynomial with a factor of r. Find a basis of Pl consisting of seven harmonic polynomials and three polynomials with a factor ofr. ... 

In four dimensions, as in three dimensions, the restrictions of homogeneous harmonic polynomials of degree n to the unit sphere are called spherical harmonic functions of degree n. The analysis in four dimensions proceeds much as it did in three dimensions, although the dimension counts change. 

Anticipating Fock s notation, let us call our variables Xi, 3 and %<. An example of a homogeneous harmonic polynomial of degree four is... 

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. 

Next, we consider the symmetry operations of the system. The free energy is expanded as a function of the strains (as defined above) and the corresponding harmonic polynomials A (a,-). The resulting expression must be invariant under the symmetry transformations. If the symmetry is low enough, one can reduce further the vector space(s) introduced above, by choosing a suitable basis. The resulting irreducible subspaces are indicated... 

For the magnetoelastic coupling parameters (B °, By-2), the first superscript indicates the irreducible representation, the second one the degree of the harmonic polynomial in (a, ). Notice that the bracketed expressions in eq. (4) can be rewritten in a form analogous to that in eq. (3a) ... 

The >+,(+ spherical harmonics which are rewritten in terms of harmonic polynomials. We now substitute (8.472), (8.473) and (8.474) into (8.470) and after some tedious but straightforward algebra succeed in regenerating equation (8.468). 

In general, this display shows how the set of harmonic polynomials for given f-value splits on Descent in Symmetry from the spherical group to the particular point group in question. The calculation is limited to -values less than or equal to 60. 

Finally we should consider the case where n = 2 and j = 0 (in physics a 3d orbital n — 3,1 — 2, m = —2, —1, l), 1,2). Now there will be no radial nodes because the Legendre polynomial is constant. All the action will be in the angular variables. When n = 2 there are several harmonic polynomials that are easy to check xy, yz, xz. It is easy to see where these are zero. 

One can rotate this picture to give the other two easy harmonic polynomials of degree 2, yz and xz. But this is not the whole story because the dimension of the space of homogeneous polynomials of degree two is 6 (given by combinations of xy, xz, yz, xx, yy, zz) and we only lose one dimension by setting... 

The construction of the Lam6 spheroconal harmonic polynomials involves Eq. (43) with matching parameters Eq. (44), matching species, and matching excitations of the respective elliptical cone coordinate degrees of freedom. The matching of species and kinds are fhe following ... 

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