steepest-descent%20paths
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21—30 of 152 matching pages
21: 28 Mathieu Functions and Hill’s Equation
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22: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
23: 23 Weierstrass Elliptic and Modular
Functions
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24: 31.9 Orthogonality
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►The integration path begins at , encircles once in the positive sense, followed by once in the positive sense, and so on, returning finally to .
The integration path is called a Pochhammer double-loop
contour (compare Figure 5.12.3).
The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning.
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►and the integration paths
, are Pochhammer double-loop contours encircling distinct pairs of singularities , , .
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►For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2).
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25: 36 Integrals with Coalescing Saddles
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26: Gergő Nemes
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►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions.
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27: Wolter Groenevelt
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►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials.
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28: 33.24 Tables
29: 8.17 Incomplete Beta Functions
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►where and the branches of and are continuous on the path and assume their principal values when .
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8.17.24
positive integers; .
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30: 11.6 Asymptotic Expansions
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