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1: Bibliography M
  • I. D. Macdonald (1968) The Theory of Groups. Clarendon Press, Oxford.
  • P. J. Mohr and B. N. Taylor (2005) CODATA recommended values of the fundamental physical constants: 2002. Rev. Mod.Phys. 77, pp. 1–107.
  • 2: 27.20 Methods of Computation: Other Number-Theoretic Functions
    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). … To compute a particular value p ( n ) it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
    3: 4.46 Tables
    Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4). … For 40D values of the first 500 roots of tan x = x , see Robinson (1972). … For 10S values of the first five complex roots of sin z = a z , cos z = a z , and cosh z = a z , for selected positive values of a , see Fettis (1976). …
    4: 28.7 Analytic Continuation of Eigenvalues
    §28.7 Analytic Continuation of Eigenvalues
    The branch points are called the exceptional values, and the other points normal values. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). All real values of q are normal values. …
    5: 28.34 Methods of Computation
    Methods available for computing the values of w I ( π ; a , ± q ) needed in (28.2.16) include:
  • (a)

    Direct numerical integration of the differential equation (28.2.1), with initial values given by (28.2.5) (§§3.7(ii), 3.7(v)).

  • §28.34(ii) Eigenvalues
  • (c)

    Solution of (28.2.1) by boundary-value methods; see §3.7(iii). This can be combined with §28.34(ii)(c).

  • (b)

    Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

  • 6: 4.31 Special Values and Limits
    §4.31 Special Values and Limits
    Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
    z 0 1 2 π i π i 3 2 π i
    7: 23.23 Tables
    2 in Abramowitz and Stegun (1964) gives values of ( z ) , ( z ) , and ζ ( z ) to 7 or 8D in the rectangular and rhombic cases, normalized so that ω 1 = 1 and ω 3 = i a (rectangular case), or ω 1 = 1 and ω 3 = 1 2 + i a (rhombic case), for a = 1. …The values are tabulated on the real and imaginary z -axes, mostly ranging from 0 to 1 or i in steps of length 0. 05, and in the case of ( z ) the user may deduce values for complex z by application of the addition theorem (23.10.1). …
    8: 5.21 Methods of Computation
    An effective way of computing Γ ( z ) in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). Or we can use forward recurrence, with an initial value obtained e. …
    9: 20.14 Methods of Computation
    The Fourier series of §20.2(i) usually converge rapidly because of the factors q ( n + 1 2 ) 2 or q n 2 , and provide a convenient way of calculating values of θ j ( z | τ ) . …For instance, the first three terms of (20.2.1) give the value of θ 1 ( 2 i | i ) ( = θ 1 ( 2 i , e π ) ) to 12 decimal places. For values of | q | near 1 the transformations of §20.7(viii) can be used to replace τ with a value that has a larger imaginary part and hence a smaller value of | q | . …In theory, starting from any value of τ , a finite number of applications of the transformations τ τ + 1 and τ 1 / τ will result in a value of τ with τ 3 / 2 ; see §23.18. In practice a value with, say, τ 1 / 2 , | q | 0.2 , is found quickly and is satisfactory for numerical evaluation.
    10: 31.18 Methods of Computation
    Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of z ; see Laĭ (1994) and Lay et al. (1998). …