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SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Spherical harmonics: a theoretical and graphical study av Christian Helanow 29 - No 8 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 69 STOCKHOLM Spherical harmonics: a theoretical and graphical study Christian Helanow Självständigt arbete i matematik 5 högskolepoäng, grundnivå Handledare: Andreas Axelsson 29 Abstract The topic of harmonic polynomials is briefly discussed to show that every polynomial on R n can be decomposed into harmonic polynomials. Using this property it is proved that every function that is square integrable on the hypersphere can be represented by a series of spherical harmonics (harmonic polynomials restricted to the hypersphere), and that the series is converging with respect to the norm in this space. Explicit formulas for these functions and series are calculated for three dimensional euclidean space and used for graphical illustrations. By applying stereographic projection a way of graphically illustrating spherical harmonics in the plane and how a given function is approximated by a sum of spherical harmonics is presented. Contents Introduction 3 2 Harmonic polynomials 4 2. Definitions and notations The orthogonal decomposition of polynomials The dimension of homogeneous harmonic polynomials Spherical Harmonics Zonal Harmonics 2 3. Zonal Harmonics in the series expansion of a given function Properties of zonal harmonics Spherical Harmonics in Spherical Coordinates 7 4. Eigenfunctions to Laplace s equation The Legendre Polynomial Oscillations of the Zonal Harmonic Solutions to Legendre s associated equation Series of Spherical Harmonics Graphical illustrations Illustrations Legendre polynomials and Zonal Harmonics Spherical Harmonics Approximation of functions in stereographic coordinates. 32 A Appendix A 4 A. The Laplace operator in Spherical Coordinates A.. The spherical Laplace operator in stereographic coordinates 4 B Appendix B 42 2 Introduction From Fourier analysis it is known that an infinite set of orthogonal sine and cosine functions span the space of square integrable functions on the interval [ π, π]. By considering functions in n-dimensional space that solve Laplace s equation, a subclass of functions called spherical harmonics can be defined. These functions can be shown to be an analogue to the sine and cosine functions in the sense that spherical harmonics of different degrees form an orthogonal basis that spans the space of functions that are square integrable on the sphere. This study is meant to present some of the available information on spherical harmonics in a way that appeals to a reader at the undergraduate level. The main aim is to establish a clear connection between the special cases of Fourier analysis in R 2 and spherical harmonics in R 3, both by using theory and by graphically illustrating the spherical harmonics in a number of ways. Each section is structured around one or a few central results. These will be introduced at the beginning of each section, in the form of a discussion or as a stated theorem. After this is done, the tools needed to prove the relevant theorems will be introduced. The purpose of this layout is to give the reader an appreciation of the importance and consequences of the central theorems. The first part of this study, presented in section 2 and 3, is concerned with the general topic of harmonic polynomials and how these can be restricted to the sphere to define spherical harmonics. Since this theoretical part is not greatly facilitated by only considering three dimensional euclidean space, it will include complex valued functions f(x) : R n C. No explicit formulas for spherical harmonics are derived in this section, which can make it seem somewhat abstract. It is recommended that the reader looks through the graphical illustrations at the end of this study to get an intuitive understanding of spherical harmonics while reading the general theory. Later sections will take a more formal approach to these illustrations. The main result of section 2 is the orthogonal decomposition of functions that are square integrable on the sphere into spherical harmonics, which is presented in Theorem 2.6. In section 3 we focus on a way to calculate the unique spherical harmonics that decompose a given function, which is given by the formula in Theorem 3.2. However, as can be seen in later sections that include explicit calculations, this formula is of theoretical rather than practical value. The second part (section 4) is theory applied to three dimensional euclidean space. The emphasis in this section is on finding the solution to Laplace s equation in spherical coordinates. The answer results in an explicit expression for spherical harmonics in three dimensions. In Theorem 4.4 it is summarized how to find the expansion of a given function into spherical harmonics. We will only consider the case of real-valued functions, thus finding formulas that can be applied directly in the final section containing the illustrations. The last part (section 5) graphically illustrates the theoretical concepts from part two. Examples will be given of both traditional ways of doing this, as well as less common ways (not found in literature during the research of this study). To accomplish this, some theory about stereographic projection is presented. The general disposition of section 2 and 3 are inspired heavily by [, Chapter 5]. Some theorems have been added or chosen to be proved in a different way. If so, their source will be referred to in the text. The idea to section 4 is from [3, Chapter ], [4] and [8]. 3 2 Harmonic polynomials 2. Definitions and notations In this study n will always denote a positive integer. A function f that is square integrable on R n is written as f L 2 (R n ). A function f(x) defined on an open subset of R n that is at least twice continuously differentiable and fulfills Laplace s equation () 2 f x f x 2, () n is called harmonic. Defining the Laplacian operator as the sum of all the second partial derivatives the above condition can be written as f. (2) Note that this definition applies to complex valued functions, since it would only mean that the real and imaginary parts of f are both harmonic. As is customary R denotes the real numbers and C the complex numbers. If a function f is continuous on a given set K, this will be denoted f C(K). The unit sphere is defined as the boundary of the unit ball, and denoted as S. It is understood that if dealing with a subset of R n, the surface that is the unit sphere has dimension n. 2.2 The orthogonal decomposition of polynomials A polynomial p(x) on R n is called homogeneous of degree k if for a constant t it fulfills p(tx) = t k p(x). The space of polynomials that are homogeneous of degree k will be denoted P k (R n ) and the subspace of P k (R n ) containing those polynomials that are harmonic will be denoted H k (R n ). Note that every polynomial P of degree k on R n can be written as P = k j= p j, where each p j P j (R n ). Since P = k j= p j, we have that P is harmonic if and only if each p j H j (R n. Given this fact, this section will focus on the polynomials p k H k (R n ). The main result of this section is about the decomposition of homogeneous polynomials. This is presented in the theorem below. Theorem 2.. If k 2, then P k (R n ) = H k (R n ) x 2 P k 2 (R n ). However, before proving the statement above, let us consider some important consequences. Theorem 2. states that every homogeneous polynomial p P k (R n ) can be decomposed in this way. Naturally this argument can be transferred to a homogeneous polynomial q P k 2 (R n ). Extending this to polynomials of lesser degree we get: p = p k + x 2 q, for some p k H k (R n ), q P k 2 (R n ), q = p k 2 + x 2 s, for some p k 2 H k 2 (R n ), s P k 4 (R n ),. 4 This relation holds k 2 times (or the largest integer less than this), leaving the last term to contain either a polynomial of degree or a constant. Substituting the above relations stepwise leads us to a corollary to Theorem 2.. Corollary 2.2. Every p P k (R n ) can be uniquely written in the form p = p k + x 2 p k x 2m p k 2m, where m denotes the largest integer less than or equal to k 2 (that is k 2m equals if k is even, if k is odd) and p j H j (R n ). Proof. The proof has already been outlined in the argument above. Noting that P k (R n ) = H k (R n ) for k =,, we see that the statement is true for these values of k. For k 2, the proof is by induction assuming that the equality holds when k is replaced by k 2. This holds because of Theorem 2., giving the above result. For the uniqueness of the decomposition, assume that p k + x 2 q k 2 = p k + x 2 q k 2, where p k, p k H k (R n ) and q k 2, q k 2 P k 2 (R n ). This is equivalent to p k p k = x 2 q k 2 x 2 q k 2. Since the left hand side of the equation above is a harmonic polynomial, this must also be true for the right hand side. But according to Theorem 2. the right hand side does not belong to H k (R n ). Hence the only way the equality can hold is if q k 2 q k 2 =. The importance of this corollary becomes apparent when considering polynomials that are restricted to the sphere. In this special case Corollary 2.2 becomes the following statement. Corollary 2.3. Any homogeneous polynomial p P k (R n ) restricted to the unit sphere can be uniquely written on the form p = p k + p k p k 2m, where m denotes the largest integer less than or equal to k 2 (that is k 2m equals if k is even, if k is odd) and p j H j (R n ). Proof. Just applying the fact that any power of the factor x 2 = on S to Corollary 2.2, gives us the decomposition of p. We will only comment on the uniqueness by observing that the decomposition is harmonic and is the (unique) solution to the Dirichlet problem for the ball (see [, p. 2]) when the boundary data is the restriction of p to the sphere. From our initial discussion about homogeneous polynomials we know that Corollary 2.3 indirectly states that any polynomial on the sphere can be written as a sum of unique harmonic homogeneous polynomials (on the sphere). Here we have already hinted at the importance of Theorem 2. and that this leads to a special reason to study harmonic polynomials on the sphere. Since this topic will be more thoroughly discussed in section 2.4, we leave this special case for now. So far we have only stated Theorem 2.. For the proof it will be necessary to rely on some facts from linear algebra about the decomposition of dual spaces. In particular the following about adjoint mappings is used ([5, p. 24]). 5 Adjoint mappings. Let E and F be inner product spaces. Then the linear map ϕ : E F induces a map ϕ : F E satisfying ϕx,y = x, ϕy, (3) where ϕ and ϕ are said to be adjoint. By this relation F can be orthogonally decomposed as F = Im ϕ ker ϕ. (4) For a thorough discussion on the definitions and the linear algebra used, see for instance [5, Chapter II]. We now prove Theorem 2. ([7, Theorem 4..]). Proof of Theorem 2.. The goal of this proof is to find adjoint maps from P k (R n ) P k 2 (R n ) and P k 2 (R n ) P k (R n ), such that equation (4) can be used to determine the orthogonal decomposition of these spaces. To accomplish this an inner product is defined to suit this specific purpose. To facilitate this process we introduce multi-index notation at this point. If x R n and α = (α,α 2,...,α n ) we define x α = x α xα2 2...xαn n, α! = α!α 2!...α n!, α = α + α α n α x α =... x α x α2 2 x αn n. and Any polynomial p(x) P k (R n ) can be written on the form p(x) = α c α x α, where α = k, and c α C. Using the operator p(d) = α c α α x α, an inner product on P k (R n ) can be defined as follows. p,q = p(d)[q] = α c α x α d β x β α =k β =k = c α d β δ αβ α!, α, β =k where p,q P k (R n ), c α and d β are (complex) constants and δ αβ = if α = β and δ αβ = if α β. Now assume that p P k (R n ) is orthogonal to x 2 P k 2 (R n ), so that x 2 q,p = 6 for all q P k 2 (R n ). Then by the definition of the inner product we get x 2 q,p = ( x 2 q)(d)[p] n = 2 α 2 x 2 cα j x α (p) j= = q(d)[p] = q(d)[ p] = q(d)[ p] = q, p =. Since p P k 2 (R n ), the calculation above indicates that p is orthogonal to every q P k 2 (R n ). This can only be true if p, and therefore p H k (R n ). Now consider the map ϕ : P k (R n ) P k 2 (R n ) such that p p. According to the above argument and equation (3) this has the adjoint map ϕ : P k 2 (R n ) P k (R n ) such that q x 2 q. With Im ϕ = { x 2 q; q P k 2 (R n )} and ker ϕ = {p; p H k (R n )}, equation (4) shows that P k (R n ) = H k (R n ) P k 2 (R n ). Finally, note that the orthogonality in Theorem 2. implies that no polynomial times the factor x 2 is harmonic. 2.3 The dimension of homogeneous harmonic polynomials This section is dedicated to finding dimh k (R n ). We start by considering the case n = 2. From complex analysis it is known that any polynomial p(z) = a +a z +a 2 z 2... (where a,a,a 2... are complex constants) can be written on the complex form p(x,y) = u(x,y)+iv(x,y). Since every polynomial p(z) is analytic it follows that u(x,y) and v(x,y) are both harmonic functions. For a homogeneous polynomial of degree k we have that u = a kz k +a k z k 2 and v = a kz k a k z k 2i. This indicates that both z k and z k are homogeneous harmonic polynomials. Hence every homogeneous harmonic polynomial p k can be written as a complex linear combination of {z k,z k }. From this we can see that dimh k (R 2 ) = 2 for all values of k. When k = we have that p is a constant function and only has dimension equal to one. For n 2, we note that Theorem 2. gives that the dimension of H k (R n ) is equal to dimp k (R n ) minus dimp k 2 (R n ). Hence all that is needed is to find dimp k (R n ). This can be accomplished through combinatorics, as is shown in the proof of the proposition below. For the values k =, every homogeneous polynomial is harmonic, so we restrict our attention to k 2. Proposition 2.4. If k 2, then ( ) ( ) n + k n + k 3 dimh k (R n ) = n n 7 Proof. If we use the multi-index notation introduced in the proof of Theorem 2., we are looking for all the unique monomials x α such that α = k. This set of combinations can be seen as a basis for the space of homogeneous polynomials of degree k, since every p P k (R n ) is a linear combination of these elements. In other words, we are asking the question What is the number of unordered selections, with repetition, of k objects from a set of n objects that can be made? ([2, Theorem.2]). The answer is found in combinatorics and is ( ) ( ) n + k n + k =. k n Hence the expression above equals dimp k (R n ). Similarly dimp k 2 (R n ) = ( ) n+k 3 n. Since dimh k (R n ) =dimp k (R n ) dimp k 2 (R n ), this finishes the proof. We can easily calculate dimh k (R n ) for n = 2, by using the formula from Theorem 2.2. ( ) ( ) k + k dimh k (R 2 ) = = (k + ) (k ) = 2 for values of k 2. This confirms our previous argument that any p H k (R 2 ) is in the complex linear span of {z k,z k }. For n = 3 a similar calculation show that dimh k (R 3 ) = 2k +, so the dimension of homogeneous harmonic polynomials increase linearly with the degree. Further calculations can be made and will reveal that if n = 4 the dimension will increase in a quadratic manner, if n = 5 in a cubic manner etc. Now that we have discussed the basic properties of homogeneous harmonic polynomials, we are ready to study what the results will be if these are restricted to the sphere. This will be the main purpose of the next section, which contains the general theory of spherical harmonics. 2.4 Spherical Harmonics In section 2.2 we concluded that the restriction of harmonic polynomials to the sphere resulted in important consequences, which motivates the following definition. Definition 2.5. A homogeneous harmonic polynomials of degree k on R n restricted to the unit sphere is called a spherical harmonic of degree k. The set of spherical harmonics of degree k is denoted H k (S n ). If the situation permits, the dimension of the sphere will be omitted and the set will be denoted H k (S). The aim of this section is to find an (infinite) orthogonal set of functions that span the space L 2 (S). That the set of spherical harmonics should be such a set can be motivated by the following argument considering Fourier analysis. Usually one thinks of a Fourier series as an expansion of a given 8 function f(x) L 2 [ π,π] into trigonometric functions on the interval [ π,π]. This is usually written as f(x) = a 2 + (a k cos kx + b k sin kx), k= where a, a k and b k are constants. However, from trigonometry we know that the sine and cosine functions are functions defined on the unit circle, and that these can be written on exponential form as cos θ = eiθ +e iθ 2 and sinθ = eiθ e iθ 2i (where θ is the angle from the positive x-axis). This gives us the above Fourier series in exponential form f(x) = k= c k e ikθ, (5) where c k are complex constants that are directly related to a k and b k from the ordinary Fourier series. From Parseval s equation and Fourier analysis we conclude that the (complex) linear span of {e ikθ,e ikθ } k= is dense in the space of L 2 [ π,π]([3], p. 9). But when θ [ π,π] the set {e ikθ,e ikθ } k= is just the restriction of {z k,z k } k= to the one dimensional subspace that is the unit circle. Hence the Fourier series in equation (5) can be seen as an expansion into spherical harmonics. Thus, in the special case n = 2, a Fourier expansion into spherical harmonics and Parseval s equation shows us that L 2 (S) = k= H k(s). This leads us to state the main theorem of this section. Theorem 2.6. The infinite set {H k (S)} k= is an orthogonal decomposition of the space L 2 (S) so that L 2 (S) = k=h k (S). As has already been shown in Corollary 2.3, any polynomial restricted to the sphere can be written as a sum of spherical harmonics. However, we want to expand the concept of spherical harmonics being an orthogonal basis in the space of polynomials restricted to the sphere to spherical harmonics being an orthogonal basis in the space L 2 (S), as is presented in Theorem 2.6. A theorem of great importance in accomplishing this task is the Stone-Weierstrass theorem (S-W). To get a first idea how this is done we will again think about the case of Fourier analysis. The S-W for one variable is as follows ([, Theorem 7.26]). If f is a continuous complex function on [a,b], there exists a sequence of polynomials P n such that lim n P n(x) = f(x) uniformly on [a,b]. Together with Corollary 2.3 and Parseval s equation, the S-W results in L 2 [ π,π] = k= H k(s). An analogous result of the S-W holds in higher dimension ([, Theorem 7.33]). Stone-Weierstrass Theorem 2.7. Suppose A is a self-adjoint algebra of complex continuous functions on a compact set K. If to every distinct pair of points x,x 2 K, there corresponds a function f A such that f(x ) f(x 2 ) (A separates points on K) and 9 to each x K there corresponds a function g A such that g(x) (A vanishes at no point of K), then A is dense in C(K). If we can confirm that the space of polynomials (A in S-W) restricted to the sphere (K in S-W) fulfills the conditions in Theorem 2.7, we can say that these are dense in the space of continuous functions restricted to the sphere. If P k (S) is the complex vector space of homogeneous polynomials of degree k restricted to the sphere, then any polynomial p k P k (S) can be written as its real and imaginary part p k = q k +is k where q k,s k P k (S). Hence p k = q k is k also belongs to P k (S) and the space of homogeneous polynomials is self-adjoint. Since every polynomial can be decomposed into a sum of homogeneous polynomials, this also applies to P(S). To show that P(S) separates any distinct points x,y S, x y, consider the set of functions {p p = x k ; k =,...n} where (x,x 2,...,x n ) is the basis
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