Addition Theorem Spherical Harmonics

4 Key excerpts on "Addition Theorem 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)

  9.3.3. Addition Theorem for Spherical Harmonics. The addition theorem between two different directions in the spherical coordinate system is given in § 9.2.3. Another useful result is as follows. Theorem 9.5. (Addition Theorem for spherical harmonics) The following relations hold: P n (cos γ ) = 4 π 2 n + 1 n m = − n ( − 1) m Y m n ( θ 1 , φ 1 ) Y − m n ( θ 2 , φ 2 ) , (9.60) or equivalently, P n (cos γ ) = 4 π 2 n + 1 n m = − n ( − 1) m Y m n ( θ 1 , φ 1 ) Y m ∗ n ( θ 2 , φ 2 ); (9.61) or in terms of the associated Legendre functions of the first kind: P n (cos γ ) = P n (cos θ 1 ) P n (cos θ 2 ) + 2 n m =1 ( n − m )! ( n + m )! P m n (cos θ 1 ) P m n (cos θ 2 ) cos m ( φ 1 − φ 2 ) . (9.62) Proof. We will derive (9.61), and other result follow. Let a function f ( θ, φ ) be expanded in a Laplace’s series f ( θ 1 , φ 1 ) =    Y m n ( θ 1 , φ 1 ) relative to x 1 , y 1 , z 1 , n ∑ m = − n a nm Y m n ( γ, ψ ) relative to x 2 , y 2 , z 2 , (9.63) 270 9. SPHERICAL HARMONICS where ψ is the azimuth angle, with any choice of the 0 of this angle (see Fig. 9.2). At γ = 0 we have f ( θ 1 , φ 1 ) γ =0 = a n 0 2 n + 1 4 π , (9.64) since P n (1) = 1 , and P m n (1) = 0 for m = 0 . Multiplying (9.63) by Y 0 ∗ n ( γ, ψ ) and integrating over the surface S of the sphere, we get S f ( θ 1 , φ 1 ) Y 0 ∗ n ( γ, ψ ) dS γ,ψ = a n 0 , which in view of (9.64) can be written as S Y m n ( θ 1 , φ 1 ) Y 0 ∗ n ( γ, ψ ) dS = a n 0 . (9.65) x 1 1 y z 1 P 1 θ 1 φ 2 θ 2 φ γ z 2 x 2 y 2 Fig. 9.2. Let us assume that the polynomial P n (cos θ ) has an expansion of the form P n (cos θ ) = n m = − n b nm Y m n ( θ 1 , φ 1 ) , (9.66) where b nm depend on θ 2 , φ 2 , i.e., on the orientation of the z 2 -axis. Multiplying the above integral by Y m ∗ n ( θ 1 , φ 1 ) and integrating with respect to θ 1 and φ 1 over S , we obtain S P n (cos γ ) Y m ∗ n ( θ 1 , φ 1 ) dS θ 1 ,φ 1 = b nm , which in terms of the spherical harmonics becomes 4 π 2 n + 1 S Y 0 n ( γ, ψ ) Y m ∗ n ( θ 1 , φ 1 ) dS = b nm , (9.67)

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  Texture Analysis in Materials Science

  Mathematical Methods

  • H.-J. Bunge(Author)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  They thus possess rotational symmetry with respect to the Z-axes of the sample fixed and crystal fixed coordinate systems, respectively. If in the addition theorem for generalized spherical harmonics (equation 14.6) one sets m = 0, one thus obtains the addition theorem for spherical surface har-monics : W > / ) = Σ Μ(Φ>7)Τ! η (9) (14.45) Φ', γ' denote the coordinates of the direction Φ, y in the coordinate system rotated through gr 1 . In particular, it follows from equation (14.8) with equation (14.40) that _ +i _ ργ(φ + Φ') = Σ i n -s Pf(0) Pf n {0') (14.46) 8=-l and Ρ ι {Φ + Φ') = |/_iL· J ^ (-1) 4 Ρ!(Φ) W ) (14.47) The spherical harmonics fulfil the orthonormalization condition φ kf(0, /?) ijK(

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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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  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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