Euler%20splines
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1: 24.1 Special Notation
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Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 24.17 Mathematical Applications
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§24.17(ii) Spline Functions
►Euler Splines
… ►The functions …are called Euler splines of degree . … ►Bernoulli Monosplines
…3: Publications
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B. V. Saunders and Q. Wang (2006)
From B-Spline Mesh Generation to Effective Visualizations for the
NIST Digital Library of Mathematical Functions,
in Curve and Surface Design, Proceedings of the Sixth International
Conference on Curves and Surfaces,
Avignon, France June 29–July 5, 2006,
pp. 235–243.
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B. Saunders and Q. Wang (2010)
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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B. I. Schneider, B. R. Miller and B. V. Saunders (2018)
NIST’s Digital Library of Mathematial Functions,
Physics Today
71, 2, 48 (2018), pp. 48–53.
4: 5.22 Tables
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►Abramowitz and Stegun (1964, Chapter 6) tabulates , , , and for to 10D; and for to 10D; , , , , , , , and for to 8–11S; for to 20S.
Zhang and Jin (1996, pp. 67–69 and 72) tabulates , , , , , , , and for to 8D or 8S; for to 51S.
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►Abramov (1960) tabulates for () , () to 6D.
Abramowitz and Stegun (1964, Chapter 6) tabulates for () , () to 12D.
…Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of , , and for , to 8S.
5: Bibliography D
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A Practical Guide to Splines.
Revised edition, Applied Mathematical Sciences, Vol. 27, Springer-Verlag, New York.
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On the real roots of Euler polynomials.
Monatsh. Math. 106 (2), pp. 115–138.
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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials.
J. Approx. Theory 49 (4), pp. 321–330.
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Zeros of Bernoulli, generalized Bernoulli and Euler polynomials.
Mem. Amer. Math. Soc. 73 (386), pp. iv+94.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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6: 24.20 Tables
§24.20 Tables
… ►Wagstaff (1978) gives complete prime factorizations of and for and , respectively. …7: Bibliography S
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Cardinal Spline Interpolation.
Society for Industrial and Applied Mathematics, Philadelphia, PA.
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Spline Functions: Basic Theory.
John Wiley & Sons Inc., New York.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective.
Numer. Math. 98 (2), pp. 371–387.
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Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities.
Math. Comp. 81 (280), pp. 2159–2173.
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8: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Zhang and Jin (1996, Table 3.8) tabulates for , to 8D or 8S.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
9: 24.2 Definitions and Generating Functions
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§24.2(ii) Euler Numbers and Polynomials
… ►§24.2(iii) Periodic Bernoulli and Euler Functions
… ► ► …10: 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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►This is the number of positive integers that are relatively prime to ; is Euler’s totient.
►If , then the Euler–Fermat theorem states that
…The numbers are relatively prime to and distinct (mod ).
…Note that .
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