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31: 10.72 Mathematical Applications
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►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)).
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►If has a double zero , or more generally is a zero of order , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order .
The number
can also be replaced by any real constant
in the sense that
is analytic and nonvanishing at ; moreover, is permitted to have a single or double pole at .
The order of the approximating Bessel functions, or modified Bessel functions, is , except in the case when has a double pole at .
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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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32: 26.1 Special Notation
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►Other notations for , the Stirling numbers of the first kind, include (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), (Jordan (1939), Moser and Wyman (1958a)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)).
►Other notations for , the Stirling numbers of the second kind, include (Fort (1948)), (Jordan (1939)), (Moser and Wyman (1958b)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
binomial coefficient. | |
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Eulerian number. | |
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Bell number. | |
Catalan number. | |
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33: 25.11 Hurwitz Zeta Function
34: 24.10 Arithmetic Properties
35: 26.12 Plane Partitions
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►An equivalent definition is that a plane partition is a finite subset of with the property that if and , then must be an element of .
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►The number of plane partitions of is denoted by , with .
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►in it is
…in it is
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►in it is
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36: 4.19 Maclaurin Series and Laurent Series
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4.19.5
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37: 24.7 Integral Representations
38: 24.6 Explicit Formulas
39: 27.2 Functions
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27.2.13
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