relations%20to%20other%20functions
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11: Bibliography W
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Elliptic Functions According to Eisenstein and Kronecker.
Classics in Mathematics, Springer-Verlag, Berlin.
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Rapid approximation to the Voigt/Faddeeva function and its derivatives.
J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.
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Some explicit Padé approximants for the function
and a related quadrature formula involving Bessel functions.
SIAM J. Math. Anal. 16 (4), pp. 887–895.
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12: 27.2 Functions
§27.2 Functions
… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►Other examples of number-theoretic functions treated in this chapter are as follows. … ►This is Jordan’s function. … ►This is Liouville’s function. …13: 18.40 Methods of Computation
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►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree.
For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev.
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►See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby.
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►The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to
and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
Results of low (
to
decimal digits) precision for are easily obtained for
to
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14: 26.10 Integer Partitions: Other Restrictions
§26.10 Integer Partitions: Other Restrictions
… ►§26.10(ii) Generating Functions
… ►where the last right-hand side is the sum over of the generating functions for partitions into distinct parts with largest part equal to . … ►§26.10(iii) Recurrence Relations
… ►§26.10(vi) Bessel-Function Expansion
…15: 36.2 Catastrophes and Canonical Integrals
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►with the contour passing to the lower right of .
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is related to the Airy function (§9.2):
…(Other notations also appear in the literature.)
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►Addendum: For further special cases see §36.2(iv)
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16: 9.9 Zeros
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and have no other zeros.
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§9.9(ii) Relation to Modulus and Phase
… ►§9.9(iii) Derivatives With Respect to
… ►§9.9(iv) Asymptotic Expansions
… ►§9.9(v) Tables
…17: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of , , and , and also to complex values. … ►§8.17(ii) Hypergeometric Representations
… ►§8.17(iv) Recurrence Relations
… ►§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties
…18: Bibliography O
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Summing one- and two-dimensional series related to the Euler series.
J. Comput. Appl. Math. 98 (2), pp. 245–271.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations.
Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.
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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions.
J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
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Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms.
SIAM J. Math. Anal. 22 (5), pp. 1475–1489.
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19: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Cyclic identities for Jacobi elliptic and related functions.
J. Math. Phys. 44 (4), pp. 1822–1841.
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Theta relations and projective normality of Abelian varieties.
Amer. J. Math. 98 (4), pp. 865–889.
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The structure relation for Askey-Wilson polynomials.
J. Comput. Appl. Math. 207 (2), pp. 214–226.
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20: 19.36 Methods of Computation
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►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i).
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►To (19.36.6) add
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►Lee (1990) compares the use of theta functions for computation of , , and , , with four other methods.
…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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