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11: 34.9 Graphical Method
§34.9 Graphical Method
… βΊFor specific examples of the graphical method of representing sums involving the , and symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).12: 34.10 Zeros
…
βΊIn a symbol, if the three angular momenta do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the symbol is zero.
…Such zeros are called nontrivial zeros.
βΊFor further information, including examples of nontrivial zeros and extensions to symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).
13: 34.13 Methods of Computation
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βΊMethods of computation for and symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981).
βΊFor symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989).
…
14: 34.7 Basic Properties: Symbol
§34.7 Basic Properties: Symbol
… βΊ§34.7(ii) Symmetry
… βΊ§34.7(iv) Orthogonality
… βΊ§34.7(vi) Sums
… βΊIt constitutes an addition theorem for the symbol. …15: 34.1 Special Notation
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βΊ
βΊ
βΊThe main functions treated in this chapter are the Wigner symbols, respectively,
…
βΊAn often used alternative to the symbol is the Clebsch–Gordan coefficient
…For other notations for , , symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).
nonnegative integers. | |
… |
16: 1.12 Continued Fractions
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βΊ
is called the th approximant or convergent to
.
and are called the th (canonical) numerator and denominator respectively.
…
βΊ
is equivalent to if there is a sequence , ,
, such that … βΊwhen , . …when , . …
, such that … βΊwhen , . …when , . …
17: 16.7 Relations to Other Functions
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βΊFor , , symbols see Chapter 34.
Further representations of special functions in terms of functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).
18: 9.4 Maclaurin Series
19: 23 Weierstrass Elliptic and Modular
Functions
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20: 10 Bessel Functions
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