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1: 21.10 Methods of Computation
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§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
… ►Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.
2: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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On the asymptotic analysis of the Painlevé equations via the isomonodromy method.
Nonlinearity 7 (5), pp. 1291–1325.
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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent.
Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).
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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution.
J. Phys. A 31 (17), pp. 4073–4113.
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3: Bibliography W
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Asymptotics of orthogonal polynomials via recurrence relations.
Anal. Appl. (Singap.) 10 (2), pp. 215–235.
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Uniform asymptotic expansion of
via a difference equation.
Numer. Math. 91 (1), pp. 147–193.
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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach.
J. Math. Pures Appl. (9) 85 (5), pp. 698–718.
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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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On the connection formulas of the fourth Painlevé transcendent.
Anal. Appl. (Singap.) 7 (4), pp. 419–448.
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4: 31.18 Methods of Computation
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►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 ; see Laĭ (1994) and Lay et al. (1998).
…The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.
5: Bibliography K
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Series expansions for the third incomplete elliptic integral via partial fraction decompositions.
J. Comput. Appl. Math. 207 (2), pp. 331–337.
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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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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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6: 20 Theta Functions
Chapter 20 Theta Functions
…7: 18.42 Software
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►Citations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web.
►A more complete list of available software for computing these functions, and for generating formulas symbolically, is found in the Software Index.
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8: 18.40 Methods of Computation
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Stieltjes Inversion via (approximate) Analytic Continuation
… ►Results of low ( to decimal digits) precision for are easily obtained for to . … ►The bottom and top of the steps at the are lower and upper bounds to as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43). … ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …where the coefficients are defined recursively via , and …9: 2.11 Remainder Terms; Stokes Phenomenon
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§2.11(ii) Connection Formulas
… ►However, on combining (2.11.6) with the connection formula (8.19.18), with , we derive … ►§2.11(vi) Direct Numerical Transformations
… ►For example, using double precision is found to agree with (2.11.31) to 13D. … ►For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.10: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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From Klein to Painlevé via Fourier, Laplace and Jimbo.
Proc. London Math. Soc. (3) 90 (1), pp. 167–208.
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