change%20of%20variables
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1: 12.11 Zeros
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§12.11(ii) Asymptotic Expansions of Large Zeros
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12.11.5
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12.11.6
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12.11.8
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12.11.9
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2: 9.7 Asymptotic Expansions
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►Numerical values of are given in Table 9.7.1 for to 2D.
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§9.7(iii) Error Bounds for Real Variables
… ►In (9.7.7) and (9.7.8) the th error term is bounded in magnitude by the first neglected term multiplied by where for (9.7.7) and for (9.7.8), provided that in the first case and in the second case. … ►§9.7(iv) Error Bounds for Complex Variables
… ►provided that , for (9.7.5) and , for (9.7.6). …3: 20.11 Generalizations and Analogs
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►In the case identities for theta functions become identities in the complex variable
, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).
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§20.11(iii) Ramanujan’s Change of Base
… ►However, in this case is no longer regarded as an independent complex variable within the unit circle, because is related to the variable of the theta functions via (20.9.2). … ►These results are called Ramanujan’s changes of base. …4: 5.11 Asymptotic Expansions
5: 12.10 Uniform Asymptotic Expansions for Large Parameter
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►In this section we give asymptotic expansions of PCFs for large values of the parameter that are uniform with respect to the variable
, when both and
are real.
These expansions follow from Olver (1959), where detailed information is also given for complex variables.
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►The variable
is defined by
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12.10.40
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12.10.45
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6: 19.36 Methods of Computation
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►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18).
…If (19.36.1) is used instead of its first five terms, then the factor in Carlson (1995, (2.2)) is changed to .
►For both and the factor in Carlson (1995, (2.18)) is changed to when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms:
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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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►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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7: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter.
SIAM J. Math. Anal. 21 (4), pp. 995–1018.
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8: 20.10 Integrals
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