absolutely convergent
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11: 20.5 Infinite Products and Related Results
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►With the given conditions the infinite series in (20.5.10)–(20.5.13) converge absolutely and uniformly in compact sets in the -plane.
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►These double products are not absolutely convergent; hence the order of the limits is important.
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12: 1.17 Integral and Series Representations of the Dirac Delta
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►for all functions that are continuous when , and for each ,
converges absolutely for all sufficiently large values of .
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►More generally, assume is piecewise continuous (§1.4(ii)) when for any finite positive real value of , and for each ,
converges absolutely for all sufficiently large values of .
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►provided that is continuous when , and for each ,
converges absolutely for all sufficiently large values of (as in the case of (1.17.6)).
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13: 25.14 Lerch’s Transcendent
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►
25.14.6
if ;
, if .
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14: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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►The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important.
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15: 1.9 Calculus of a Complex Variable
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►The series converges absolutely if
converges.
A series
converges (diverges) absolutely when (), or when ().
Absolutely convergent series are also convergent.
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►The double series is absolutely convergent if it is convergent when is replaced by .
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►If a double series is absolutely convergent, then it is also convergent and its sum is given by either of the repeated sums
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16: 15.2 Definitions and Analytical Properties
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►
(a)
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Converges absolutely when .
17: 16.2 Definition and Analytic Properties
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►On the circle the series (16.2.1) is absolutely convergent if , convergent except at if , and divergent if , where
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18: 21.2 Definitions
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►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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19: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
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►The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the -plane.
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