Glaisher constant
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1: 5.17 Barnes’ -Function (Double Gamma Function)
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5.17.5
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►Here is the Bernoulli number (§24.2(i)), and is Glaisher’s constant, given by
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5.17.6
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5.17.7
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►For Glaisher’s constant see also Greene and Knuth (1982, p. 100) and §2.10(i).
2: 22.1 Special Notation
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►The notation , , is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882).
Other notations for are and with ; see Abramowitz and Stegun (1964) and Walker (1996).
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3: 27.21 Tables
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►Glaisher (1940) contains four tables: Table I tabulates, for all : (a) the canonical factorization of into powers of primes; (b) the Euler totient ; (c) the divisor function ; (d) the sum of these divisors.
…7 of Abramowitz and Stegun (1964) also lists the factorizations in Glaisher’s Table I(a); Table 24.
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4: 2.10 Sums and Sequences
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►Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1).
…where is Euler’s constant (§5.2(ii)) and is the derivative of the Riemann zeta function (§25.2(i)).
is sometimes called Glaisher’s constant.
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►where () is a real constant, and
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►for any real constant
and the set of all positive integers , we derive
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5: 22.4 Periods, Poles, and Zeros
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§22.4(ii) Graphical Interpretation via Glaisher’s Notation
…6: 3.12 Mathematical Constants
§3.12 Mathematical Constants
►The fundamental constant …Other constants that appear in the DLMF include the base of natural logarithms …see §4.2(ii), and Euler’s constant … ►For access to online high-precision numerical values of mathematical constants see Sloane (2003). …7: 30.1 Special Notation
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►The main functions treated in this chapter are the eigenvalues and the spheroidal wave functions , , , , and , .
…Meixner and Schäfke (1954) use , , , for , , , , respectively.
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►Flammer (1957) and Abramowitz and Stegun (1964) use for , for , and
…where is a normalization constant determined by
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real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, . |
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arbitrary small positive constant. |
8: 16.15 Integral Representations and Integrals
9: 32.9 Other Elementary Solutions
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►with , , , and arbitrary constants.
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►with an arbitrary constant, which is solvable by quadrature.
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►with and arbitrary constants.
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►with an arbitrary constant, which is solvable by quadrature.
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►with and arbitrary constants.
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