Hölder’s
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21: 20.8 Watson’s Expansions
§20.8 Watson’s Expansions
…22: 31.17 Physical Applications
§31.17 Physical Applications
►§31.17(i) Addition of Three Quantum Spins
… ►We use vector notation (respective scalar ) for any one of the three spin operators (respective spin values). ►Consider the following spectral problem on the sphere : . … ►For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …23: 31.6 Path-Multiplicative Solutions
§31.6 Path-Multiplicative Solutions
►A further extension of the notation (31.4.1) and (31.4.3) is given by ►
31.6.1
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►with , but with another set of .
This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the -plane that encircles and once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor .
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24: 29.14 Orthogonality
25: 28.17 Stability as
26: 25.2 Definition and Expansions
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►When ,
…Elsewhere is defined by analytic continuation.
It is a meromorphic function whose only singularity in is a simple pole at , with residue 1.
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►This includes, for example, .
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►product over zeros of with (see §25.10(i)); is Euler’s constant (§5.2(ii)).
27: 25.5 Integral Representations
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►Throughout this subsection .
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25.5.5
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25.5.13
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►In (25.5.15)–(25.5.19), , is the digamma function, and is Euler’s constant (§5.2).
(25.5.16) is also valid for , .
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28: 25.15 Dirichlet -functions
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►If , then is an entire function of .
…This implies that if .
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►Since if , (25.15.5) shows that for a primitive character the only zeros of for (the so-called trivial zeros) are as follows:
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►This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11).
Related results are:
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29: 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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Stirling numbers of the first kind. | |
Stirling numbers of the second kind. |
30: 25.19 Tables
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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of for both real and complex . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of , and §22.17 lists tables for some Dirichlet -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.