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11—20 of 608 matching pages
11: 8 Incomplete Gamma and Related
Functions
Chapter 8 Incomplete Gamma and Related Functions
…12: 1.11 Zeros of Polynomials
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►Set to reduce to , with , .
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, , , .
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►Resolvent cubic is with roots , , , and , , .
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►Let
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►Then , with , is stable iff ; , ; , .
13: 19.37 Tables
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►Tabulated for , to 10D by Fettis and Caslin (1964).
►Tabulated for , to 7S by Beli͡akov et al. (1962).
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►Tabulated for , to 10D by Fettis and Caslin (1964).
►Tabulated for , to 6D by Byrd and Friedman (1971), for , and to 8D by Abramowitz and Stegun (1964, Chapter 17), and for , to 9D by Zhang and Jin (1996, pp. 674–675).
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►Tabulated for , , to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)).
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14: 21.5 Modular Transformations
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►Let , , , and be matrices with integer elements such that
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21.5.1
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►Here is an eighth root of unity, that is, .
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►( invertible with integer elements.)
…For a matrix we define , as a column vector with the diagonal entries as elements.
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15: 3.3 Interpolation
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►If is analytic in a simply-connected domain (§1.13(i)), then for ,
…where is a simple closed contour in described in the positive rotational sense and enclosing the points .
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►If is analytic in a simply-connected domain , then for ,
…where is given by (3.3.3), and is a simple closed contour in described in the positive rotational sense and enclosing .
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►Then by using in Newton’s interpolation formula, evaluating and recomputing , another application of Newton’s rule with starting value gives the approximation , with 8 correct digits.
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16: Bibliography K
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Linear convergence and the bisection algorithm.
Amer. Math. Monthly 93 (1), pp. 48–51.
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Special functions and the Bieberbach conjecture.
Amer. Math. Monthly 95 (8), pp. 689–696.
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Calculation of the complex zeros of the modified Bessel function of the second kind and its derivatives.
Zh. Vychisl. Mat. i Mat. Fiz. 24 (8), pp. 1150–1163.
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The addition formula for Laguerre polynomials.
SIAM J. Math. Anal. 8 (3), pp. 535–540.
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Bessel Functions and their Applications.
Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.
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17: Bibliography N
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Generalization of Binet’s Gamma function formulas.
Integral Transforms Spec. Funct. 24 (8), pp. 597–606.
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Der Eulersche Dilogarithmus und seine Verallgemeinerungen.
Nova Acta Leopoldina 90, pp. 123–212.
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Mémoire sur les polynomes de Bernoulli.
Acta Math. 43, pp. 121–196 (French).
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The asymptotic behavior of the general real solution of the third Painlevé equation.
Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).
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