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11: Errata
  • Section 14.30

    In regard to the definition of the spherical harmonics Y l , m , the domain of the integer m originally written as 0 m l has been replaced with the more general | m | l . Because of this change, in the sentence just below (14.30.2), “tesseral for m < l and sectorial for m = l ” has been replaced with “tesseral for | m | < l and sectorial for | m | = l ”. Furthermore, in (14.30.4), m has been replaced with | m | .

    Reported by Ching-Li Chai on 2019-10-05

  • 12: 1.7 Inequalities
    §1.7(iii) Means
    1.7.7 H G A ,
    13: 1.17 Integral and Series Representations of the Dirac Delta
    Spherical Harmonics14.30)
    1.17.25 δ ( cos θ 1 - cos θ 2 ) δ ( ϕ 1 - ϕ 2 ) = = 0 m = - Y , m ( θ 1 , ϕ 1 ) Y , m ( θ 2 , ϕ 2 ) ¯ .
    14: 18.39 Physical Applications
    For a harmonic oscillator, the potential energy is given by …
    15: 1.9 Calculus of a Complex Variable
    Harmonic Functions
    Mean Value Property
    For u ( z ) harmonic, …
    Poisson Integral
    is harmonic in | z | < R . …
    16: Bibliography T
  • A. Terras (1988) Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.
  • O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.
  • T. Ton-That, K. I. Gross, D. St. P. Richards, and P. J. Sally (Eds.) (1995) Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.
  • 17: 1.10 Functions of a Complex Variable
    Harmonic Functions
    If u ( z ) is harmonic in D , z 0 D , and u ( z ) u ( z 0 ) for all z D , then u ( z ) is constant in D . Moreover, if D is bounded and u ( z ) is continuous on D ¯ and harmonic in D , then u ( z ) is maximum at some point on D . …
    18: Bibliography G
  • W. Gautschi (1974) A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.
  • A. Gil and J. Segura (1998) A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.
  • A. Gil and J. Segura (2000) Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.
  • A. Gil and J. Segura (2001) DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.
  • 19: Bibliography H
  • E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.
  • L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.
  • 20: Bibliography W
  • G. Weiss (1965) Harmonic Analysis. In Studies in Real and Complex Analysis, I. I. Hirschman (Ed.), Studies in Mathematics, Vol. 3, pp. 124–178.
  • E. T. Whittaker (1902) On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.