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21: 3.11 Approximation Techniques
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►Beginning with , , we apply
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►With , the last equations give as the solution of a system of linear equations.
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►(3.11.29) is a system of linear equations for the coefficients .
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►With this choice of and , the corresponding sum (3.11.32) vanishes.
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►Two are endpoints: and ; the other points and are control points.
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22: 21.1 Special Notation
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positive integers. | |
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th element of vector . | |
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Transpose of . | |
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set of all elements of the form “”. | |
set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).) | |
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23: Bibliography K
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Evaluation of complex zeros of Bessel functions and and their derivatives.
Zh. Vychisl. Mat. i Mat. Fiz. 24 (10), pp. 1497–1513.
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The asymptotic expansion of a hypergeometric function
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Math. Comp. 26 (120), pp. 963.
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An extension of Saalschütz’s summation theorem for the series
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Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
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On the complex zeros of for real or complex order.
J. Comput. Appl. Math. 40 (3), pp. 337–344.
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Some special cases of the generalized hypergeometric function
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J. Comput. Appl. Math. 78 (1), pp. 79–95.
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24: 19.29 Reduction of General Elliptic Integrals
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►Let
…where
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►Next, for , define , and assume both ’s are positive for .
…where
…If , where both linear factors are positive for , and , then (19.29.25) is modified so that
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25: 4.17 Special Values and Limits
26: Bibliography E
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The penetration of a potential barrier by electrons.
Phys. Rev. 35 (11), pp. 1303–1309.
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Painlevé transcendent describes quantum correlation function of the antiferromagnet away from the free-fermion point.
J. Phys. A 29 (17), pp. 5619–5626.
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Institutiones Calculi Integralis.
Opera Omnia (1), Vol. 11, pp. 110–113.
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The Sobolev orthogonality and spectral analysis of the Laguerre polynomials for positive integers
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J. Comput. Appl. Math. 171 (1-2), pp. 199–234.
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Note on the -Laguerre orthogonal polynomials.
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27: 24.19 Methods of Computation
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►Equations (24.5.3) and (24.5.4) enable and to be computed by recurrence.
…For example, the tangent numbers can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied.
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►For other information see Chellali (1988) and Zhang and Jin (1996, pp. 1–11).
For algorithms for computing , , , and see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).
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§24.19(ii) Values of Modulo
…28: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into at most distinct parts.
…The set is denoted by .
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►It is known that for , , with strict inequality for sufficiently large, provided that , or ; see Yee (2004).
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►where is the modified Bessel function (§10.25(ii)), and
…The quantity is real-valued.
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29: Bibliography O
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Studies on the Painlevé equations. III. Second and fourth Painlevé equations, and
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Math. Ann. 275 (2), pp. 221–255.
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Studies on the Painlevé equations. I. Sixth Painlevé equation
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Ann. Mat. Pura Appl. (4) 146, pp. 337–381.
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Studies on the Painlevé equations. II. Fifth Painlevé equation
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Japan. J. Math. (N.S.) 13 (1), pp. 47–76.
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Studies on the Painlevé equations. IV. Third Painlevé equation
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Funkcial. Ekvac. 30 (2-3), pp. 305–332.
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Numerical solution of Riemann-Hilbert problems: Painlevé II.
Found. Comput. Math. 11 (2), pp. 153–179.
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30: 3.9 Acceleration of Convergence
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►A transformation of a convergent sequence with limit into a sequence is called limit-preserving if converges to the same limit .
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►This transformation is accelerating if is a linearly convergent
sequence, i.
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►Then the transformation of the sequence into a sequence is given by
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►Then .
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►We give a special form of Levin’s transformation in which the sequence of partial sums is transformed into:
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