Pochhammer%20integral
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11: 17.13 Integrals
§17.13 Integrals
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17.13.1
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17.13.2
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Ramanujan’s Integrals
… ►Askey (1980) conjectured extensions of the foregoing integrals that are closely related to Macdonald (1982). …12: 17.2 Calculus
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►For ,
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►For properties of the function see §27.14.
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§17.2(v) Integrals
►If is continuous at , then …13: 18.25 Wilson Class: Definitions
14: 19.19 Taylor and Related Series
§19.19 Taylor and Related Series
… ►The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23): … ►Then has at most one term if in the series for . For and , has at most one term if , and two terms if or 5. … ►15: 19.5 Maclaurin and Related Expansions
§19.5 Maclaurin and Related Expansions
… ► … ►An infinite series for is equivalent to the infinite product … ►Series expansions of and are surveyed and improved in Van de Vel (1969), and the case of is summarized in Gautschi (1975, §1.3.2). …16: 8.20 Asymptotic Expansions of
§8.20 Asymptotic Expansions of
►§8.20(i) Large
… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). ►For an exponentially-improved asymptotic expansion of see §2.11(iii). ►§8.20(ii) Large
…17: 17.6 Function
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17.6.14
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§17.6(v) Integral Representations
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17.6.28
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17.6.29
►where , , and the contour of integration separates the poles of from those of , and the infimum of the distances of the poles from the contour is positive.
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18: 18.28 Askey–Wilson Class
19: 18.27 -Hahn Class
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►In case of unbounded sequences (18.27.2) can be rewritten as a -integral, see §17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)).
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18.27.14_1
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18.27.16
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18.27.19
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18.27.20
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