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11: 34.4 Definition: Symbol
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34.4.1
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►Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths ; see Figure 34.4.1.
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34.4.2
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►where is defined as in §16.2.
►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
12: 1.3 Determinants, Linear Operators, and Spectral Expansions
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►For real-valued ,
…for every distinct pair of , or when one of the factors vanishes.
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►where are the th roots of unity (1.11.21).
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►Let be defined for all integer values of and , and denote the determinant
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►The spectrum of such self-adjoint operators consists of their eigenvalues, , and all .
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13: 3.4 Differentiation
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►The are the differentiated Lagrangian interpolation coefficients:
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►where and .
►For the values of and used in the formulas below
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►For partial derivatives we use the notation .
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14: 34.3 Basic Properties: Symbol
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►When any one of is equal to , or , the symbol has a simple algebraic form.
…For these and other results, and also cases in which any one of is or , see Edmonds (1974, pp. 125–127).
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►Even permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,
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34.3.8
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►For the polynomials see §18.3, and for the function see §14.30.
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15: 19.29 Reduction of General Elliptic Integrals
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►The Cauchy principal value is taken when or is real and negative.
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►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as multiplied either by or by in the cases of (19.29.20) or (19.29.21), respectively.
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►Next, for , define , and assume both ’s are positive for .
…If , where both linear factors are positive for , and , then (19.29.25) is modified so that
…In the cubic case, in which , , (19.29.26) reduces further to
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16: 5.10 Continued Fractions
17: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
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►It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
…Note that .
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►In the following examples, are the exponents in the factorization of in (27.2.1).
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►Table 27.2.1 lists the first 100 prime numbers .
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18: 23.9 Laurent and Other Power Series
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►Let be the nearest lattice point to the origin, and define
…Explicit coefficients in terms of and are given up to in Abramowitz and Stegun (1964, p. 636).
►For , and with as in §23.3(i),
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►where , if either or , and
…For with and , see Abramowitz and Stegun (1964, p. 637).
19: 26.16 Multiset Permutations
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►Let be the multiset that has copies of , .
denotes the set of permutations of for all distinct orderings of the integers.
The number of elements in is the multinomial coefficient (§26.4) .
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►The
-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by
…and again with we have
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20: 34.2 Definition: Symbol
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►The quantities in the symbol are called angular momenta.
…where is any permutation of .
The corresponding projective quantum numbers
are given by
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►where is defined as in §16.2.
►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).