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1: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into parts with difference at least 3, except that multiples of 3 must differ by at least 6.
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►where the sum is over nonnegative integer values of for which .
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►where the sum is over nonnegative integer values of for which .
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►where the sum is over nonnegative integer values of for which .
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►The quantity is real-valued.
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2: 34.2 Definition: Symbol
§34.2 Definition: Symbol
►The quantities in the symbol are called angular momenta. …They therefore satisfy the triangle conditions …where is any permutation of . The corresponding projective quantum numbers are given by …3: 10.75 Tables
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Makinouchi (1966) tabulates all values of and in the interval , with at least 29S. These are for , 10, 20; , ; with and , except for .
British Association for the Advancement of Science (1937) tabulates , , , 7–8D; , , , 7–10D; , , , , , 8D. Also included are auxiliary functions to facilitate interpolation of the tables of , for small values of .
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
Olver (1960) tabulates , , , , , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as .
4: 6.16 Mathematical Applications
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6.16.1
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►The first maximum of for positive occurs at
and equals ; compare Figure 6.3.2.
Hence if and , then the limiting value of overshoots by approximately 18%.
Similarly if , then the limiting value of undershoots by approximately 10%, and so on.
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5: 19.36 Methods of Computation
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►Complex values of the variables are allowed, with some restrictions in the case of that are sufficient but not always necessary.
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►Accurate values of for near 0 can be obtained from by (19.2.6) and (19.25.13).
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►This method loses significant figures in if and are nearly equal unless they are given exact values—as they can be for tables.
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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►When the values of complete integrals are known, addition theorems with (§19.11(ii)) ease the computation of functions such as when is small and positive.
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6: 25.12 Polylogarithms
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►Other notations and names for include (Kölbig et al. (1970)), Spence function (’t Hooft and Veltman (1979)), and (Maximon (2003)).
►In the complex plane has a branch point at
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►For other values of , is defined by analytic continuation.
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7: 28.16 Asymptotic Expansions for Large
8: 27.20 Methods of Computation: Other Number-Theoretic Functions
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►To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2).
For a completely multiplicative function we use the values at the primes together with (27.3.10).
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►To compute a particular value
it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9).
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►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function , and the values can be checked by the congruence (27.14.20).
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