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1: Bibliography M
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The Theory of Groups.
Clarendon Press, Oxford.
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CODATA recommended values of the fundamental physical constants: 2002.
Rev. Mod.Phys. 77, pp. 1–107.
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2: 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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3: 4.46 Tables
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►Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4).
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►For 40D values of the first 500 roots of , see Robinson (1972).
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►For 10S values of the first five complex roots of , , and , for selected positive values of , see Fettis (1976).
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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 are normal values. … ► …5: 28.34 Methods of Computation
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►Methods available for computing the values of needed in (28.2.16) include:
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(a)
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(c)
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(b)
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§28.34(ii) Eigenvalues
… ►6: 4.31 Special Values and Limits
§4.31 Special Values and Limits
► …7: 23.23 Tables
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►2 in Abramowitz and Stegun (1964) gives values of , , and to 7 or 8D in the rectangular and rhombic cases, normalized so that and (rectangular case), or and (rhombic case), for = 1.
…The values are tabulated on the real and imaginary -axes, mostly ranging from 0 to 1 or in steps of length 0.
05, and in the case of the user may deduce values for complex by application of the addition theorem (23.10.1).
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8: 5.21 Methods of Computation
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►An effective way of computing 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.
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9: 20.14 Methods of Computation
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►The Fourier series of §20.2(i) usually converge rapidly because of the factors or , and provide a convenient way of calculating values of .
…For instance, the first three terms of (20.2.1) give the value of () to 12 decimal places.
►For values of near the transformations of §20.7(viii) can be used to replace with a value that has a larger imaginary part and hence a smaller value of .
…In theory, starting from any value of , a finite number of applications of the transformations and will result in a value of with ; see §23.18.
In practice a value with, say, , , is found quickly and is satisfactory for numerical evaluation.