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sectorial harmonics

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11: 1.7 Inequalities
§1.7(iii) Means
1.7.7 H G A ,
12: 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 ) ¯ .
13: 18.38 Mathematical Applications
Zonal Spherical Harmonics
Ultraspherical polynomials are zonal spherical harmonics. …
14: 18.39 Applications in the Physical Sciences
argument a) The Harmonic Oscillator … This is illustrated in Figure 18.39.1 where the first and fourth excited state eigenfunctions of the Schrödinger operator with the rationally extended harmonic potential, of (18.39.19), are shown, and compared with the first and fourth excited states of the harmonic oscillator eigenfunctions of (18.39.14) of paragraph a), above. … The eigenfunctions of L 2 are the spherical harmonics Y l , m l ( θ , ϕ ) with eigenvalues 2 l ( l + 1 ) , each with degeneracy 2 l + 1 as m l = l , l + 1 , , l . … …
18.39.24 Ψ n , l , m l ( r , θ , ϕ ) = R n , l ( r ) Y l , m l ( θ , ϕ ) .
15: 25.11 Hurwitz Zeta Function
25.11.32 0 a x n ψ ( x ) d x = ( 1 ) n 1 ζ ( n ) + ( 1 ) n H n B n + 1 n + 1 k = 0 n ( 1 ) k ( n k ) H k B k + 1 ( a ) k + 1 a n k + k = 0 n ( 1 ) k ( n k ) ζ ( k , a ) a n k , n = 1 , 2 , , a > 0 ,
where H n are the harmonic numbers:
25.11.33 H n = k = 1 n k 1 .
16: 1.2 Elementary Algebra
§1.2(iv) Means
The geometric mean G and harmonic mean H of n positive numbers a 1 , a 2 , , a n are given by …
1.2.19 1 H = 1 n ( 1 a 1 + 1 a 2 + + 1 a n ) .
M ( 1 ) = H ,
17: 1.9 Calculus of a Complex Variable
Harmonic Functions
Mean Value Property
For u ( z ) harmonic, …
Poisson Integral
is harmonic in | z | < R . …
18: 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.
  • 19: 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.
  • D. Gómez-Ullate and R. Milson (2014) Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47 (1), pp. 015203, 26 pp..
  • 20: 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 . …