double sequence
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1: 1.9 Calculus of a Complex Variable
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§1.9(vii) Inversion of Limits
►Double Sequences and Series
►A set of complex numbers where and take all positive integer values is called a double sequence. … ►A double series is the limit of the double sequence … …2: Guide to Searching the DLMF
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phrase:
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any double-quoted sequence of textual words and numbers.
3: Bibliography C
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Algorithm 597: Sequence of modified Bessel functions of the first kind.
ACM Trans. Math. Software 9 (2), pp. 242–245.
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4: 3.8 Nonlinear Equations
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►Let be a sequence of approximations to a root, or fixed point, .
…for all sufficiently large, where and are independent of , then the sequence is said to have convergence of the
th order.
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►For real functions the sequence of approximations to a real zero will always converge (and converge quadratically) if either:
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►As in the case of Table 3.8.1 the quadratic nature of convergence is clearly evident: as the zero is approached, the number of correct decimal places doubles at each iteration.
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►Starting this iteration in the neighborhood of one of the four zeros , sequences
are generated that converge to these zeros.
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5: 5.17 Barnes’ -Function (Double Gamma Function)
6: Bibliography W
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Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL.
ACM Trans. Math. Software 25 (2), pp. 240–250.
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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series.
Computer Physics Reports 10 (5-6), pp. 189–371.
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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations.
Computers in Physics 10 (5), pp. 496–503.
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Sequence Transformations and their Applications.
Mathematics in Science and Engineering, Vol. 154, Academic Press Inc., New York.
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On the central connection problem for the double confluent Heun equation.
Math. Nachr. 195, pp. 267–276.
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7: 5.4 Special Values and Extrema
8: 18.33 Polynomials Orthogonal on the Unit Circle
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►This states that for any sequence
with and the polynomials generated by the recurrence relations (18.33.23), (18.33.24) with satisfy the orthogonality relation (18.33.17) for a unique probability measure with infinite support on the unit circle.
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18.33.32
9: 4.2 Definitions
10: Bibliography S
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Coulomb functions analytic in the energy.
Comput. Phys. Comm. 25 (1), pp. 87–95.
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Non-linear transformations of divergent and slowly convergent sequences.
J. Math. Phys. 34, pp. 1–42.
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Algorithm AS 239. Chi-squared and incomplete gamma integral.
Appl. Statist. 37 (3), pp. 466–473.
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On the double points of a Mathieu equation.
J. Comput. Appl. Math. 107 (1), pp. 111–125.
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The On-Line Encyclopedia of Integer Sequences.
Notices Amer. Math. Soc. 50 (8), pp. 912–915.
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