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21—30 of 670 matching pages
21: 26.14 Permutations: Order Notation
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βΊAs an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
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βΊThe permutation has two descents: and .
…For example, .
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βΊIn this subsection is again the Stirling number of the second kind (§26.8), and is the th Bernoulli number (§24.2(i)).
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βΊ
26.14.11
.
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22: 1.3 Determinants, Linear Operators, and Spectral Expansions
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βΊSquare matices can be seen as linear operators because for all and , the space of all -dimensional vectors.
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βΊThe adjoint of a matrix is the matrix such that for all .
In the case of a real matrix and in the complex case .
βΊReal symmetric () and Hermitian () matrices are self-adjoint operators on .
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βΊFor self-adjoint and , if , see (1.2.66), simultaneous eigenvectors of and always exist.
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23: 25.16 Mathematical Applications
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βΊ
25.16.2
,
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βΊwhere is given by (25.11.33).
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βΊ
has a simple pole with residue () at each odd negative integer , .
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βΊ
25.16.14
βΊ
25.16.15
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24: 11.11 Asymptotic Expansions of Anger–Weber Functions
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βΊLet , and for ,
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βΊ
.
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βΊWhen is real and positive, all of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–360) for as , one being uniform for , and the other being uniform for .
(Note that Olver’s definition of omits the factor in (11.10.4).)
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βΊLastly, corresponding asymptotic approximations and expansions for and , with or , follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions and ; see §10.19(ii).
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25: Bibliography M
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Formulas and Theorems for the Special Functions of Mathematical Physics.
3rd edition, Springer-Verlag, New York-Berlin.
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On the Representation of Meijer’s -Function in the Vicinity of Singular Unity.
In Complex Analysis and Applications ’81 (Varna, 1981),
pp. 383–398.
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On the roots of the Bessel and certain related functions.
Ann. of Math. 9 (1-6), pp. 23–30.
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Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion.
J. Lond. Math. Soc. 9, pp. 6–13 (German).
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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26: Bibliography B
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Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy.
Appl. Math. Lett. 9 (5), pp. 21–26.
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Avoided crossings of the quartic oscillator.
J. Phys. A 30 (9), pp. 3057–3067.
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Methods of calculation of radial wave functions and new tables of Coulomb functions.
Physical Rev. (2) 80, pp. 553–560.
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Algorithm 524: MP, A Fortran multiple-precision arithmetic package [A1].
ACM Trans. Math. Software 4 (1), pp. 71–81.
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The determination of phases and amplitudes of wave functions.
Proc. Phys. Soc. 81 (3), pp. 442–452.
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27: 15.7 Continued Fractions
28: 25.6 Integer Arguments
29: 22.17 Moduli Outside the Interval [0,1]
30: 3.5 Quadrature
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βΊIf , then the remainder in (3.5.2) can be expanded in the form
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βΊFor the Bernoulli numbers see §24.2(i).
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βΊAbout function evaluations are needed.
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βΊwith weight function
, is one for which whenever is a polynomial of degree .
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βΊFor further information, see Mason and Handscomb (2003, Chapter 8), Davis and Rabinowitz (1984, pp. 74–92), and Clenshaw and Curtis (1960).
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