via divided differences
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11—20 of 21 matching pages
11: 9.13 Generalized Airy Functions
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►In Olver (1977a, 1978) a different normalization is used.
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►The function on the right-hand side is recessive in the sector , and is therefore an essential member of any numerically satisfactory pair of solutions in this region.
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►When is not an integer the branch of in (9.13.25) is usually chosen to be with .
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►and the difference equation
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►For further generalizations via integral representations see Chin and Hedstrom (1978), Janson et al. (1993, §10), and Kamimoto (1998).
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12: 24.17 Mathematical Applications
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Calculus of Finite Differences
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24.17.7
,
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►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); -adic analysis (Koblitz (1984, Chapter 2)).
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13: 16.17 Definition
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►Then the Meijer
-function is defined via the Mellin–Barnes integral representation:
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(i)
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(ii)
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(iii)
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►Assume , no two of the bottom parameters , , differ by an integer, and is not a positive integer when and .
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goes from to . The integral converges if and .
is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all () if , and for if .
is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all if , and for if .
14: Bibliography W
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Asymptotics of orthogonal polynomials via recurrence relations.
Anal. Appl. (Singap.) 10 (2), pp. 215–235.
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Uniform asymptotic expansion of
via a difference equation.
Numer. Math. 91 (1), pp. 147–193.
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Asymptotic expansions for second-order linear difference equations with a turning point.
Numer. Math. 94 (1), pp. 147–194.
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Linear difference equations with transition points.
Math. Comp. 74 (250), pp. 629–653.
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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach.
J. Math. Pures Appl. (9) 85 (5), pp. 698–718.
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15: 16.8 Differential Equations
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►When no is an integer, and no two
differ by an integer, a fundamental set of solutions of (16.8.3) is given by
…For other values of the , series solutions in powers of (possibly involving also ) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations.
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►When , and no two
differ by an integer, another fundamental set of solutions of (16.8.3) is given by
…More generally if () is an arbitrary constant, , and , then
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►When and some of the
differ by an integer a limiting process can again be applied.
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16: 11.10 Anger–Weber Functions
17: 1.2 Elementary Algebra
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►For complex the binomial coefficient is defined via (1.2.6).
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►A vector of norm unity is normalized and every non-zero vector can be normalized via
.
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►The difference between and is the commutator denoted as
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►Non-defective matrices are precisely the matrices which can be diagonalized via a similarity transformation of the form
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►The matrix exponential is defined via
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18: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms.
J. Comput. Phys. 100 (2), pp. 294–296.
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Liouville-Green approximations via the Riccati transformation.
J. Math. Anal. Appl. 116 (1), pp. 147–165.
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A Survey on the Liouville-Green (WKB) Approximation for Linear Difference Equations of the Second Order.
In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.),
pp. 567–577.
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A stable quotient-difference algorithm.
Math. Comp. 34 (150), pp. 515–519.
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19: 25.2 Definition and Expansions
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25.2.4
►where the Stieltjes constants are defined via
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►This includes, for example, .
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25.2.12
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