convergence properties
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1: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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Convergence Properties
…2: 3.3 Interpolation
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§3.3(vi) Other Interpolation Methods
… ►These references also describe convergence properties of the interpolation formulas. …3: 3.10 Continued Fractions
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►can be converted into a continued fraction of type (3.10.1), and with the property that the th convergent
to is equal to the th partial sum of the series in (3.10.3), that is,
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4: 2.5 Mellin Transform Methods
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►when this integral converges.
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►To ensure that the integral (2.5.3) converges we assume that
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►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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►The extended transform has the same properties as in the half-plane .
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►Next from Table 2.5.1 we observe that the integrals for the transform pair and are absolutely convergent in the domain specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20).
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5: 16.2 Definition and Analytic Properties
§16.2 Definition and Analytic Properties
… ►§16.2(ii) Case
… ►§16.2(iii) Case
… ►§16.2(iv) Case
… ►§16.2(v) Behavior with Respect to Parameters
…6: 21.2 Definitions
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21.2.1
►This -tuple Fourier series converges absolutely and uniformly on compact sets of the and spaces; hence is an analytic function of (each element of) and (each element of) .
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7: 1.3 Determinants, Linear Operators, and Spectral Expansions
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§1.3(i) Determinants: Elementary Properties
… ►Relationships Between Determinants
… ► … ►converges (§1.9(vii)). …Hill-type determinants always converge. …8: 2.1 Definitions and Elementary Properties
§2.1 Definitions and Elementary Properties
… ►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … ►The asymptotic property may also hold uniformly with respect to parameters. … ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. … ►Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. …9: 35.2 Laplace Transform
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►where the integration variable ranges over the space .
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►Then (35.2.1) converges absolutely on the region , and is a complex analytic function of all elements of .
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►Assume that
converges, and also that its limit as is .
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10: 15.2 Definitions and Analytical Properties
§15.2 Definitions and Analytical Properties
… ►On the circle of convergence, , the Gauss series: ►Converges absolutely when .