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1: 19.2 Definitions
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►where is a polynomial in while and are rational functions of .
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►Here are real parameters, and and are real or complex variables, with , .
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§19.2(iv) A Related Function:
… ►Formulas involving that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using . … ►For the special cases of and see (19.6.15). …2: 34.6 Definition: Symbol
3: 34.7 Basic Properties: Symbol
4: 26.9 Integer Partitions: Restricted Number and Part Size
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denotes the number of partitions of into at most parts.
See Table 26.9.1.
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►It follows that also equals the number of partitions of into parts that are less than or equal to .
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is the number of partitions of into at most parts, each less than or equal to .
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5: 4.17 Special Values and Limits
6: 28.6 Expansions for Small
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►Leading terms of the power series for and for are:
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►The coefficients of the power series of , and also , are the same until the terms in and , respectively.
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►Numerical values of the radii of convergence of the power series (28.6.1)–(28.6.14) for are given in Table 28.6.1.
Here for , for , and for and .
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§28.6(ii) Functions and
…7: 24.20 Tables
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►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D.
►Wagstaff (1978) gives complete prime factorizations of and for and , respectively.
In Wagstaff (2002) these results are extended to and , respectively, with further complete and partial factorizations listed up to and , respectively.
►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).
8: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
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►It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
…Note that .
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►In the following examples, are the exponents in the factorization of in (27.2.1).
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►Table 27.2.1 lists the first 100 prime numbers .
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9: 34.1 Special Notation
10: 1.3 Determinants, Linear Operators, and Spectral Expansions
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►For real-valued ,
…for every distinct pair of , or when one of the factors vanishes.
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►where are the th roots of unity (1.11.21).
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►Let be defined for all integer values of and , and denote the determinant
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►The spectrum of such self-adjoint operators consists of their eigenvalues, , and all .
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