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1: 20 Theta Functions

Chapter 20 Theta Functions

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2: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. … ►Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …

3: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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Document

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§33.22(iv).

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Double-precision Fortran code.

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1st item, 4th item.

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Error bounds for the method of steepest descents. Proc. Roy. Soc. London Ser. A 440, pp. 493–518.

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ISSN 0962-8444, FullText, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§2.3(iii), §2.4(iii), §9.10(ii).

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Gamma function asymptotics by an extension of the method of steepest descents. Proc. Roy. Soc. London Ser. A 447, pp. 609–630.

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ISSN 0962-8444, FullText, MathReview (Jens Bolte), ZentralBlatt

Referenced by:

§5.11(i), §5.11(ii), §5.11(ii), (5.9.11_1), (5.9.11_2).

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Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.

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ISSN 1073-2772, Pdf, MathReview (Dan Polis̆evski), ZentralBlatt

Referenced by:

§2.4(iv).

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4: 3.8 Nonlinear Equations

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Bisection Method

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Secant Method

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Eigenvalue Methods

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

5: Bibliography P

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

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ISSN 0010-4655, Document, Code

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6th item.

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Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.

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ISSN 0006-3835, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(vii).

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A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

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ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

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ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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6: 5.21 Methods of Computation

§5.21 Methods of Computation

… ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …

7: Bibliography C

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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An extension of the method of steepest descents. Proc. Cambridge Philos. Soc. 53, pp. 599–611.

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MathReview (P. Henrici), ZentralBlatt

Referenced by:

§2.4(v).

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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5th item.

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Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y..

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MathReview (J. B. Diaz), ZentralBlatt

Referenced by:

(14.30.8_5).

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8: 9.17 Methods of Computation

§9.17 Methods of Computation

… ►The former reference includes a parallelized version of the method. … ►In these cases boundary-value methods need to be used instead; see §3.7(iii). … ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …For the second method see also Gautschi (2002a). …

9: Publications

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Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007.

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Tensor Product B-Spline Mesh Generation for Accurate Surface Visualizations in the NIST Digital Library of Mathematical Functions, in Mathematical Methods for Curves and Surfaces, Proceedings of the 2008 International Conference on Mathematical Methods for Curves and Surfaces (MMCS 2008), Lecture Notes in Computer Science, Vol. 5862, (M. Dæhlen, M. Floater., T. Lyche, J. L. Merrien, K. Mørken, L. L. Schumaker, eds), Springer, Berlin, Heidelberg (2010) pp. 385–393.

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NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.

10: 8 Incomplete Gamma and Related
Functions

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