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1: 25.21 Software
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§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals
…2: 34.9 Graphical Method
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►For an account of this method see Brink and Satchler (1993, Chapter VII).
For 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).
3: 33.23 Methods of Computation
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►Inside the turning points, that is, when , there can be a loss of precision by a factor of approximately .
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►Noble (2004) obtains double-precision accuracy for for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7).
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§33.23(vii) WKBJ Approximations
… ► (12) should be ). … (16) should be ). …4: 10.74 Methods of Computation
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►And since there are no error terms they could, in theory, be used for all values of ; however, there may be severe cancellation when is not large compared with .
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►For evaluation of the Hankel functions and for complex values of and based on the integral representations (10.9.18) see Remenets (1973).
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►Suppose, for example, , and .
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►Methods for obtaining initial approximations to the zeros include asymptotic expansions (§§10.21(vi)-10.21(ix)), graphical intersection of graphs in (e.
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§10.74(vii) Integrals
…5: Bibliography I
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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines.
Z. Angew. Math. Mech. 75 (12), pp. 917–926.
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Highly Oscillatory Quadrature: The Story So Far.
In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.),
pp. 97–118.
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-Hermite polynomials, biorthogonal rational functions, and -beta integrals.
Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
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Classical and Quantum Orthogonal Polynomials in One Variable.
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
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From Gauss to Painlevé: A Modern Theory of Special Functions.
Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.
6: 18.30 Associated OP’s
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►and then for consecutive
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►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995).
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►For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12).
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§18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials
… ►7: 18.16 Zeros
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►except when .
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►For , and with as in §18.16(ii),
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►when .
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►All zeros of lie in the open interval .
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§18.16(vii) Discriminants
…8: 8.21 Generalized Sine and Cosine Integrals
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►Furthermore, and are entire functions of , and and are meromorphic functions of with simple poles at and , respectively.
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►When (and when , in the case of , or , in the case of ) the principal values of , , , and are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)).
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§8.21(vii) Auxiliary Functions
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8.21.24
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9: Bibliography T
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Asymptotic estimates of Stirling numbers.
Stud. Appl. Math. 89 (3), pp. 233–243.
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High Speed Numerical Integration of Fermi Dirac Integrals.
Master’s Thesis, Naval Postgraduate School, Monterey, CA.
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The Theory of Functions.
2nd edition, Oxford University Press, Oxford.
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Numerical Linear Algebra.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Rational Chebyshev approximation for the Fermi-Dirac integral
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Solid–State Electronics 41 (5), pp. 771–773.
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10: 1.10 Functions of a Complex Variable
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►If , analytic in , equals on an arc in , or on just an infinite number of points with a limit point in , then they are equal throughout and is called an analytic continuation of .
We write , to signify this continuation.
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►Suppose is analytic in the annulus
, , and .
Then
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