harmonic functions
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21—30 of 33 matching pages
21: 18.38 Mathematical Applications
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Complex Function Theory
… ►Zonal Spherical Harmonics
►Ultraspherical polynomials are zonal spherical harmonics. … ► … ►Non-Classical Weight Functions
…22: 18.39 Applications in the Physical Sciences
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► a) The Harmonic Oscillator
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►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.
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►The eigenfunctions of are the spherical harmonics
with eigenvalues , each with degeneracy as .
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c) Spherical Radial Coulomb Wave Functions
…23: Bibliography F
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Theory and Computation of Spheroidal Harmonics with General Arguments.
Master’s Thesis, The University of Western Australia, Department of Physics.
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24: Bibliography T
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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions.
Report TW 183/78
Mathematisch Centrum, Amsterdam, Afdeling Toegepaste
Wiskunde.
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Harmonic Analysis on Symmetric Spaces and Applications. II.
Springer-Verlag, Berlin.
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Hyperspherical elliptic harmonics and their relation to the Heun equation.
Phys. Rev. A 63 (032510), pp. 1–8.
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Representation Theory and Harmonic Analysis.
Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.
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Special functions in phase space: Mathieu functions.
J. Phys. A 31 (31), pp. 6725–6739.
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25: Bibliography B
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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions.
Cambridge University Press, Cambridge.
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Automatic computation of zeros of Bessel functions and other special functions.
SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.
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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V.
Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
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Associated Legendre polynomials, ordinary and modified spherical harmonics.
Comput. Phys. Comm. 5 (5), pp. 390–394.
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Two cold atoms in a harmonic trap.
Found. Phys. 28 (4), pp. 549–559.
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26: Bibliography L
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Further inequalities for the gamma function.
Math. Comp. 42 (166), pp. 597–600.
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Computations of spheroidal harmonics with complex arguments: A review with an algorithm.
Phys. Rev. E 58 (5), pp. 6792–6806.
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Monotonicity of the differences of zeros of Bessel functions as a function of order.
Proc. Amer. Math. Soc. 15 (1), pp. 91–96.
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Integral representation of the Hankel function in terms of parabolic cylinder functions.
Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.
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Expansion of the confluent hypergeometric function in series of Bessel functions.
Math. Tables Aids Comput. 13 (68), pp. 261–271.
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27: 32.2 Differential Equations
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►in which , , , , and are locally analytic functions.
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►For arbitrary values of the parameters , , , and , the general solutions of – are transcendental, that is, they cannot be expressed in closed-form elementary functions.
However, for special values of the parameters, equations – have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF.
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►When this is a nonlinear harmonic oscillator.
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28: 10.73 Physical Applications
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►With the spherical harmonic
defined as in §14.30(i), the solutions are of the form with , , , or , depending on the boundary conditions.
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§10.73(iii) Kelvin Functions
… ►§10.73(iv) Bickley Functions
…29: Bibliography V
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Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions.
CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en
Informatica, Amsterdam.
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Expansion of vacuum magnetic fields in toroidal harmonics.
Comput. Phys. Comm. 81 (1-2), pp. 74–90.
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An infinite series of Weber’s parabolic cylinder functions.
Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
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Transformations of some Gauss hypergeometric functions.
J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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Special Functions and the Theory of Group Representations.
American Mathematical Society, Providence, RI.
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30: Bibliography D
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Tables of Higher Mathematical Functions I.
Principia Press, Bloomington, Indiana.
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The constrained quantum mechanical harmonic oscillator.
Proc. Cambridge Philos. Soc. 62, pp. 277–286.
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Integral representations for elliptic functions.
J. Math. Anal. Appl. 316 (1), pp. 142–160.
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Bernoulli Numbers and Confluent Hypergeometric Functions.
In Number Theory for the Millennium, I (Urbana, IL, 2000),
pp. 343–363.
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Functions with given moments and weight functions for orthogonal polynomials.
Rocky Mountain J. Math. 23, pp. 87–104.
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