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1: 36.4 Bifurcation Sets
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§36.4(i) Formulas
►Critical Points for Cuspoids
… ►Critical Points for Umbilics
… ►This is the codimension-one surface in space where critical points coalesce, satisfying (36.4.1) and … ►This is the codimension-one surface in space where critical points coalesce, satisfying (36.4.2) and …2: 20 Theta Functions
Chapter 20 Theta Functions
…3: Foreword
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D. R. Lide (ed.), A Century of Excellence in Measurement, Standards, and Technology,
CRC Press, 2001. The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions.
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►November 20, 2009
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4: 36.5 Stokes Sets
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►Stokes sets are surfaces (codimension one) in space, across which or acquires an exponentially-small asymptotic contribution (in ), associated with a complex critical point of or .
…where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
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►the intersection lines with the bifurcation set are generated by , .
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►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets.
…The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions.
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5: 8 Incomplete Gamma and Related
Functions
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6: 28 Mathieu Functions and Hill’s Equation
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7: 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.
8: 23 Weierstrass Elliptic and Modular
Functions
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9: 36 Integrals with Coalescing Saddles
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10: 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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