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1: 19.2 Definitions
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►where is a polynomial in while and are rational functions of .
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►Here are real parameters, and and are real or complex variables, with , .
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►If , then is pure imaginary.
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§19.2(iv) A Related Function:
… ►For the special cases of and see (19.6.15). …2: Bibliography F
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Tablicy značeniĭ funkcii ot kompleksnogo argumenta.
Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).
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Application of the -function theory of Painlevé equations to random matrices: , , the LUE, JUE, and CUE.
Comm. Pure Appl. Math. 55 (6), pp. 679–727.
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Application of the -function theory of Painlevé equations to random matrices: , the JUE, CyUE, cJUE and scaled limits.
Nagoya Math. J. 174, pp. 29–114.
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Error bounds for a uniform asymptotic expansion of the Legendre function
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SIAM J. Math. Anal. 21 (2), pp. 523–535.
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Fast computation of incomplete elliptic integral of first kind by half argument transformation.
Numer. Math. 116 (4), pp. 687–719.
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3: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Chebyshev series for the spherical Bessel function
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Comput. Phys. Comm. 18 (1), pp. 73–86.
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Real zeros of hypergeometric polynomials.
J. Comput. Appl. Math. 247, pp. 152–161.
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Bessel functions and of integer order and complex argument.
Comput. Phys. Comm. 78 (1-2), pp. 181–189.
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Uniform asymptotic approximations for the Whittaker functions and
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Anal. Appl. (Singap.) 1 (2), pp. 199–212.
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4: Bibliography G
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Exponential integral for large values of
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J. Res. Nat. Bur. Standards 62, pp. 123–125.
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Some integrals involving three modified Bessel functions. I.
J. Math. Phys. 27 (3), pp. 682–687.
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Computing the conical function
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SIAM J. Sci. Comput. 31 (3), pp. 1716–1741.
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Table of
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Quart. J. Mech. Appl. Math. 1 (1), pp. 319–326.
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Von Staudt for
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Duke Math. J. 45 (4), pp. 885–910.
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5: Bibliography S
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Note on the complex zeros of
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J. Comput. Appl. Math. 201 (1), pp. 3–7.
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Recursive evaluation of - and - coefficients.
Comput. Phys. Comm. 11 (2), pp. 269–278.
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Numerical evaluation of integrals of the form and the tabulation of the function
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Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.
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Structure of avoided crossings for eigenvalues related to equations of Heun’s class.
J. Phys. A 30 (2), pp. 673–687.
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On the roots of
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Quart. Appl. Math. 46 (1), pp. 105–107.
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6: 34.6 Definition: Symbol
7: 34.5 Basic Properties: Symbol
8: 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).
9: 34.7 Basic Properties: Symbol
10: 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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