change%20of%20modulus
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1: 12.11 Zeros
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►When these zeros are the same as the zeros of the complementary error function ; compare (12.7.5).
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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: 19.36 Methods of Computation
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►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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►(In Legendre’s notation the modulus
approaches 0 or 1.)
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►The cases and require different treatment for numerical purposes, and again precautions are needed to avoid cancellations.
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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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3: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►Gauss and Legendre conjectured that is asymptotic to as :
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27.2.3
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27.2.4
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27.2.14
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4: 9.7 Asymptotic Expansions
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9.7.1
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►Numerical values of are given in Table 9.7.1 for to 2D.
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9.7.5
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9.7.6
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►where .
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5: 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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6: Errata
7: 20.11 Generalizations and Analogs
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20.11.2
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20.11.3
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§20.11(iii) Ramanujan’s Change of Base
►As in §20.11(ii), the modulus of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in -series via (20.9.1). … ►These results are called Ramanujan’s changes of base. …8: 5.11 Asymptotic Expansions
9: 3.8 Nonlinear Equations
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(a)
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(b)
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►Then the sensitivity of a simple zero to changes in is given by
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►Consider and .
We have and .
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and , do not change sign between and (monotonic convergence).
, , do not change sign in the interval , and (monotonic convergence after the first iteration).