Spherical Harmonics

9 Key excerpts on "Spherical Harmonics"

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  Green's Functions and Linear Differential Equations

  Theory, Applications, and Computation

  • Prem K. Kythe(Author)
  • 2011(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  9 Spherical Harmonics We will start with a short historical account of the development of the Spherical Harmonics. Laplace’s equation in a sphere was solved by Bernoulli’s separation method in § 4.3.3 and § 8.10. The Spherical Harmonics are the angular portion ( θ and φ portions of the spherical coordinates) of a set of solutions of Laplace’s equation. These harmonics are useful in many theoretical and physical applications, namely, in physics, seismology, geodesy, spectral analysis, magnetic fields, quantum mechanics and others. A detailed account of various approaches to Spherical Harmonics can be found in Courant and Hilbert [1968] and MacRobert [1967]. 9.1. Historical Sketch The development of this subject started with Laplace’s own account of a function Y m n ( θ, φ ) as a set of Spherical Harmonics that form an orthogonal system. This special research was developed by Laplace in 1782 in connection with the Newtonian potential for the law of universal gravitation in R 3 , when he determined that the gravitational potential P ( x ) associated with a set of point-masses m i located at points x i ∈ R 3 is defined by P ( x ) = i m i | x − x i | , where each term in this summation is a Newtonian potential at the respective point mass. About the same time Legendre had determined that the expansion for the Newtonian potential in powers of r = | x | and r 1 = | x 1 | is given by | x − x 1 | − 1 = P 0 (cos γ ) 1 r 1 + P 1 (cos γ ) r r 2 1 + P 2 (cos γ ) r 2 r 3 1 + · · · , where γ is the angle between x and x 1 , and P n are the well known Legendre polyno-mials, which are also a special case of Spherical Harmonics. The name ‘solid Spherical Harmonics’ was introduced in 1867 by William Thomson (Lord Kelvin) and Peter G. Tait in their book Treatise on Natural Philosophy to describe these functions which 251 252 9. Spherical Harmonics are the solution of homogeneous Laplace’s equation ∇ 2 u = 0 in the sphere.

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  Introduction to the Fast Multipole Method

  Topics in Computational Biophysics, Theory, and Implementation

  • Victor Anisimov, James J.P. Stewart(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 4 ; at present, it is necessary to establish a number of useful relations for Spherical Harmonics.

    3.2 Orthogonality and Normalization of Spherical Harmonics

  Spherical Harmonics Y lm (θ ,φ ) form a complete set of orthonormal, that is, orthogonal and normalized, functions. A set of functions is regarded as complete when there cannot exist another function in the coordinate space (θ ,φ ) that would be simultaneously orthogonal to all currently existing functions in the set. A complete set of spherical harmonic functions consists of an infinite number of functions, all of which are defined in the domain (θ ,φ ), and have unique pairs of indices l and m . Based on completeness of the set, any function f (θ ,φ ) defined on the surface of a unit sphere can be expanded in a series of Spherical Harmonics:

  f ( θ , ϕ ) = ∑ l = 0 ∞ ∑ m = − l l A l m Y l m ( θ , ϕ ) ,	(3.17)

  where A lm are the coefficients of expansion to be determined. In general, the index l in Equation 3.17 goes from zero to infinity. No problem arises as a result of the infinite number of members in the sum because this series quickly converges, and only a finite number of summation elements is required to obtain a convergent solution. Equation 3.17 reads as a function f (θ ,φ ) being expanded in the basis of spherical harmonic functions; therefore, it is customary to call the expansion functions basis functions .

  The completeness of the function set relies on the fact that Spherical Harmonics are orthogonal functions in the indices l and m . This means that the integral over a product of two spherical harmonic functions vanishes unless the indices of both functions are equal to each other:

  ∫ Y l ′ m ′ * ( θ , ϕ ) Y l m ( θ , ϕ ) d Ω = δ l ′ l δ m ′ m ,	(3.18)

  where d Ω = d (cos θ ) d φ indicates integration over the surface of the unit sphere. Since Spherical Harmonics are complex functions, the integral in Equation 3.18 requires one of the functions to be in its complex-conjugate form; that is, indicated by the star. Kronecker’s symbol δ l ′ l

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  Coordinates

  • Wolfram Neutsch(Author)
  • 2011(Publication Date)
  • De Gruyter

    (Publisher)

  866 23. Spherical Surface Functions 23. Spherical Surface Functions 23.1. Spherical Harmonics The simplest functions in the Euclidean vector space IR n are the polynomials in the natural coordinates. The polynomials of a fixed degree form a class of functions which is invariant under all linear transformations of IR n . Accordingly, the restrictions of the polynomials to the (n-l)-dimensional standard sphere fi c IR n are of eminent importance for approximation (inter-n polation) and numerical integration (quadrature) of real functions on sphe-rical domains. We need a few pieces of notation. Definition 23.1.1: An n-dimensional spherical (surface) function is a map f: Ω —> [R which can η be represented as the restriction of some polynomial Ρ e Pol(R ). Any such Ρ is called a representative of f. The degree of f is the minimum of the degrees of all representatives. Hence, spherical functions are always analytical. Since the unit sphere = J χ = (x , . . . ,x I 1 ) 6 R |x| 2 = x 2 + ... + χ is defined by an algebraic equation, namely Ε = 1, where we have set E(x) = Ε (χ) = Ε (χ χ ) = χ 2 + ... + χ 2 η η 1 η 1 π for the Euclidean form on R n , every spherical surface function has an infi-nite set of representatives. If Ρ is one of them, all others are contained in the general formula 23.1. Spherical Harmonics 867 Ρ* = Ρ + (Έ -1)-Q η with Q 6 PoKIR ). This imposes the question whether it is possible to dis-* tinguish a particular Ρ in a mathematically meaningful way. This can in-deed be done, as the following theorem shows: Theorem 23.1.1: (a) Among all representatives of a spherical function f: Ω —> IR, there is η exactly one which is harmonic. We call it the standard or principal re-presentative of f and denote it by the symbol H f . (b) The (algebraic) degree of H f is equal to the degree of f. Proof: We redeem the promise given in section 22. 2 and argue with the help of Di-richlet's principle. To do so, we select an arbitrary representative Ρ of f and set k = deg(P).

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  Geomathematics

  Modelling and Solving Mathematical Problems in Geodesy and Geophysics

  • Volker Michel(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  The expansion coefficients are spherical averages which reflect a pure degree dependence, which can here also be interpreted as a frequency dependence. Indeed, there are several versions of an uncertainty principle on the 192 Basis Functions sphere, which tell us that perfect space and frequency localization are mutually exclusive. For further details (including also the more general case of a d -dimensional sphere), see Dai and Xu (2014); Dang et al. (2017); Freeden et al. (1998, 2018); Iglewska-Nowak (2016); Narcowich and Ward (1996). In this respect, Spherical Harmonics represent one extremal case in the multitude of spherical trial functions which have been developed so far: they enable a perfect frequency localization, but they completely lack a space localization. The different systems of trial functions have already been categorized according to their different properties and ways of their construction. We will introduce these terminologies here. We first assume that we seek to approximate functions in a Banach space H () ⊂ L 2 () with k F k L 2 () ≤ ck F k H () for all F ∈ H () and a fixed constant c ∈ R + . We consider a function system { F k } k ∈κ in H () which is supposed to be a basis. Either H () is finite dimensional, that is, there is N ∈ N such that κ = {1, . . . , N }, or κ = N (note that the functions Y n, j can be rearranged such that they have a single index of positive integers). In the former case, this means that N = dim H () and the functions F k are linearly independent. In the latter case, we have the linear independence of every finite subsystem of { F k } k ∈κ and the property span{ F k } k ∈κ k·k H () = H (). An example of a finite-dimensional space is H () = Harm 0... d () for an arbitrary d ∈ N 0 with k·k H () = k · k L 2 () . A simple infinite-dimensional case is L 2 () itself.

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  Boundary and Eigenvalue Problems in Mathematical Physics

  • Hans Sagan(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  ϕ = 0.)

  RECOMMENDED SUPPLEMENTARY READING

  R. Courant and D. Hubert: Methods of Mathematical Physics, Vol. 1, Interscience Publishers, New York, 1953.

  E. W. Hobson: The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, London, 1931.

  T. M. MacRobert: Spherical Harmonics, Dover Publications, New York, 1948.

  Ph. Frank and R. von Mises: Die Differential und Integralgleichungen der Mechanik und Physik, Vols. I and II, F. Vieweg und Sohn, Braunschweig, 1930, 1935.

  1 This seems to be justified by experiments.

  2 This condition simply asserts that the electron is somewhere in space.

  3 See T. M. MacRobert: Spherical Harmonics, Dover Publications, New York, 1948.

  4 The permissibility of interchange of integration and summation here and on the following pages is not discussed. We refer the reader to our discussion on p. 320.

  5 See Ph. Frank and R. von Mises: Die Differential und Integralgleichungen der Mechanik und Physik, Vol. 1, F. Vieweg und Sohn, Braunschweig, 1930, pp. 748, 749.

  See also problem VIII.31 .

  6 See T. M. MacRobert: Spherical Harmonics

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  Quantum Mechanics with Applications

  • David B Beard, George B Beard, George B Beard, Kevin B Beard(Authors)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  VIII

  ELEMENTARY THREE-DIMENSIONAL WAVE FUNCTIONS IN SPHERICAL COORDINATES

  In the previous chapters we restricted our attention to problems in one dimension, because of their mathematical simplicity. Relatively few problems can be treated realistically with this simplification, however, and to understand and be able to solve most of the basic and even elementary problems in atomic and nuclear physics it is necessary to pay attention to the analysis of problems in three dimensions. In doing so we will immediately encounter differential equations different from any we have met previously, with solutions in terms of new functions which we must derive. The solutions for the fundamental bound-particle problems treated in this chapter (the rigid rotator, the spherically symmetric square well, the harmonic oscillator, and Coulomb potentials) are simple polynomials, which we will derive in detail because of their recurring importance in the study of many other more complex problems. Indeed, these polynomial functions are the basic wave functions for the analysis of the more complicated problems.

  8.1 GENERAL METHOD FOR SPHERICALLY SYMMETRIC POTENTIALS

  A great many problems in physics, having to do with the atom, the nucleus, or nuclear interactions, for example, may be calculated exactly by a spherically symmetric potential, that is, by a potential function V(r) which depends only on r and does not involve any angle variables. For this important class of problems, the three-dimensional time-independent wave function may be separated into three product wave functions,

  The time-independent Schrödinger wave equation in three dimensions is in this case

  where ∇2 , the Laplacian operator in spherical coordinates, is given by

           We proceed as in Chapter I , treating the radiation in a rectangular box, and substitute Eqs. (8.3 , 8.1 ) into Eq. (8.2 ) and multiply (8.2 ) by 1/ψ, to find

  On multiplying all terms in Eq. (8.4 ) by r2 sin2 θ and placing the φ-dependent term on the right, we find that while the left-hand side contains no φ dependence, the right-hand side contains no r or θ

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  Fourier Acoustics

  Sound Radiation and Nearfield Acoustical Holography

  • Earl G. Williams(Author)
  • 1999(Publication Date)
  • Academic Press

    (Publisher)

  m = 0 generates the recurrence relationships for the Legendre polynomials.

  6.3.3 Spherical Harmonics

  Equation 6.20 above defined the Spherical Harmonics as

  Due to Eq. (6.31) we have

  (6.44)

  The Spherical Harmonics Y n m are orthonormal:

  (6.45)

  Earlier we learned that for any complete set of orthonormal functions U n (ζ;) there exists a completeness or closure relation given by

  (6.46)

  Applying this equation to the Spherical Harmonics the completeness relation becomes

  (6.47)

  The importance of the Spherical Harmonics rests in the fact that any arbitrary function on a sphere g (θ, ϕ ) can be expanded in terms of them,

  (6.48)

  where A nm are complex constants. Because of the orthonormality of these functions the arbitrary constants can be found from

  (6.49)

  where ω is the solid angle defined by The delta function in spherical coordinates is given by

  (6.50)

  where It is easily verified that this delta function satisfies the relation

  The two-dimensional delta function on a sphere is simply which integrates to unity over the solid angle. The following is a table of some of the Spherical Harmonics.

  (6.51)

  (6.52)

  Note that for m = 0,

  (6.53)

  The simplicity of the Spherical Harmonics is borne out in the gray-scale plot of the n = 8 terms, shown in Fig. 6.5 . In this plot the values of Re[Y n m (θ, ϕ )] (m = 0, 1, ···, 8) on a unit sphere are projected onto the (y, z ) plane, looking down the positive x axis, using a gray scale with white the most positive and black the most negative in the mapping of the value of the function. The nodal lines are drawn in along with the outline of the sphere. The gray background outside the sphere is the color mapped to zero. The beauty and simplicity of the Spherical Harmonics is illustrated very well in this kind of plot. Note that Y 8 0 has no longitudinal nodal lines, whereas Y 8 1

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  Special Functions and Orthogonal Polynomials

  • Richard Beals, Roderick Wong(Authors)
  • 2016(Publication Date)
  • Cambridge University Press

    (Publisher)

  11 Spherical functions Spherical functions are solutions of the equation (1 − x 2 ) u  (x) − 2xu  (x) +  ν (ν + 1) − m 2 1 − x 2  u(x) = 0, (11.0.1) which arises from separating variables in Laplace’s equation u = 0 in spherical coordinates. Surface harmonics are the restrictions to the unit sphere of harmonic functions (solutions of Laplace’s equation) in three variables. For surface harmonics, m and ν are nonnegative integers. The case m = 0 is Legendre’s equation (1 − x 2 ) u  (x) − 2xu  (x) + ν (ν + 1) u(x) = 0, (11.0.2) with ν a nonnegative integer. The solutions to (11.0.1) that satisfy the asso- ciated boundary conditions are Legendre polynomials and certain multiples of their derivatives. These functions are the building blocks for all surface harmonics. They satisfy a number of important identities. For general values of the parameter ν , Legendre’s equation (11.0.2) has linearly independent solutions P ν (z), holomorphic for z in the complement of ( − ∞, −1], and Q ν (z), holomorphic in the complement of ( − ∞, 1]. These Legendre functions satisfy a number of identities and have several representations as integrals. For most values of the parameter ν , the four functions P ν (z), P ν (−z), Q ν (z), Q −ν−1 (z) are distinct solutions of (11.0.2), so there are linear relations connecting any three. Integer and half-integer values of ν are exceptional cases. The solutions of the spherical harmonic equation (11.0.1) with m = 1, 2, ... , are known as associated Legendre functions. They are closely related to derivatives P (m) ν and Q (m) ν of the Legendre functions. The associated Legendre functions also satisfy a number of important identities and have representations as integrals. 279 280 Spherical functions 11.1 Harmonic polynomials and surface harmonics A polynomial P(x, y, z) in three variables is said to be homogeneous of degree n if it is a linear combination of monomials of degree n: P(x, y, z) =  j+k+l=n a jkl x j y k z l .

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  Random Fields on the Sphere

  Representation, Limit Theorems and Cosmological Applications

  • Domenico Marinucci, Giovanni Peccati(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  . 3.4 Spherical Harmonics and Fourier analysis on S 2 65 (in particular, P l0 = P l ). The following inverse relations hold: D l 0m (·, ϑ, ϕ) =  4π 2l + 1 Y l−m (ϑ, ϕ) = (−1) m  4π 2l + 1 Y lm (ϑ, ϕ) . (3.37) Remark 3.25 (On notation.) In the sequel, we shall use the same notation Y lm to denote Spherical Harmonics as a function of both the spherical and the Euclidean coordinates, i.e. for all x ∈ S 2 , we shall write (with a slight abuse of notation) Y lm ( x) := Y lm (ϑ, ϕ) , x = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) . We shall also use dσ( x) to denote the Lebesgue measure on the sphere, which, in spherical coordinates is defined as dσ( x) := sin ϑdϑdϕ, meaning that, for every bounded function on S 2 ,  S 2 f ( x)dσ( x) =  2π 0  π 0 f (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) sin ϑdϑdϕ Using the content of Remark 3.24, it is immediate to check that Y l0 (ϑ, ϕ) =  2l + 1 4π P l (cos ϑ) , for all ϕ , and because P l (1) ≡ 1, for all l = 1, 2, ... Y l0 (0, 0) =  2l + 1 4π for all l = 0, 1, 2, ... . 3.4.2 Some properties of Spherical Harmonics In the sequel, for any x, y ∈ S 2 we shall label d( x, y) := arccos( x, y) the usual spherical distance, i.e. the angle between x, y, where ., . denotes the Euclidean inner product  x, y :=  x, y R 3 = x 1 y 1 + x 2 y 2 + x 3 y 3 (3.38) = sin ϑ x sin ϑ y  cos ϕ x cos ϕ y + sin ϕ x sin ϕ y  + cos ϑ x cos ϑ y = sin ϑ x sin ϑ y  cos  ϕ x − ϕ y  + cos ϑ x cos ϑ y , x = (sin ϑ x cos ϕ x , sin ϑ x sin ϕ x , cos ϑ x ) , y = (sin ϑ y cos ϕ y , sin ϑ y sin ϕ y , cos ϑ y ) . In view of (3.35) and (3.37), it is straightforward to derive a set of properties of the Spherical Harmonics, which we shall exploit heavily in the sequel of this book.

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