defined by contour integrals
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11: 3.3 Interpolation
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βΊwhere is a simple closed contour in described in the positive rotational sense and enclosing the points .
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βΊand are the Lagrangian interpolation coefficients defined by
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βΊLet be defined by
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βΊwhere 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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βΊFor interpolation of a bounded function on the cardinal
function of is defined by
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12: 2.5 Mellin Transform Methods
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βΊThe Mellin transform of is defined by
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βΊWe now apply (2.5.5) with , and then translate the integration contour to the right.
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βΊFirst, we introduce the truncated functions and
defined by
…With these definitions and the conditions (2.5.17)–(2.5.20) the Mellin transforms converge absolutely and define analytic functions in the half-planes shown in Table 2.5.1.
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βΊFor examples in which the integral defining the Mellin transform does not exist for any value of , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).
13: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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On the reciprocal modulus relation for elliptic integrals.
SIAM J. Math. Anal. 1 (4), pp. 524–526.
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Euler sums and contour integral representations.
Experiment. Math. 7 (1), pp. 15–35.
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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series.
Chelsea Publishing Co., New York.
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Numerical evaluation of the elliptic integral of the third kind.
Math. Comp. 19 (91), pp. 494–496.
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14: 12.14 The Function
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βΊthe branch of being zero when and defined by continuity elsewhere.
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§12.14(vi) Integral Representations
βΊThese follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument and parameter . … βΊHere is as in §12.10(ii), is defined by … βΊwhere is defined in (12.14.5), and (0), , (0), and are real. …15: 16.2 Definition and Analytic Properties
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βΊWhen the series (16.2.1) converges for all finite values of and defines an entire function.
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βΊIf none of the is a nonpositive integer, then the radius of convergence of the series (16.2.1) is , and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to .
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βΊSee §16.5 for the definition of as a contour integral when and none of the is a nonpositive integer.
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