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1: 3.5 Quadrature
If we add - 1 and 1 to this set of x k , then the resulting closed formula is the frequently-used ClenshawCurtis formula, whose weights are positive and given by … For detailed comparisons of the ClenshawCurtis formula with Gauss quadrature (§3.5(v)), see Trefethen (2008, 2011). … A comparison of several methods, including an extension of the ClenshawCurtis formula3.5(iv)), is given in Evans and Webster (1999). …
2: 33.23 Methods of Computation
Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … A set of consistent second-order WKBJ formulas is given by Burgess (1963: in Eq. …
3: Bibliography R
  • A. Ralston (1965) Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.
  • I. S. Reed, D. W. Tufts, X. Yu, T. K. Truong, M. T. Shih, and X. Yin (1990) Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
  • K. Reinsch and W. Raab (2000) Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.
  • H. Rosengren (1999) Another proof of the triple sum formula for Wigner 9 j -symbols. J. Math. Phys. 40 (12), pp. 6689–6691.
  • 4: Bibliography C
  • C. W. Clenshaw and A. R. Curtis (1960) A method for numerical integration on an automatic copmputer. Numer. Math. 2 (4), pp. 197–205.
  • C. W. Clenshaw, D. W. Lozier, F. W. J. Olver, and P. R. Turner (1986) Generalized exponential and logarithmic functions. Comput. Math. Appl. Part B 12 (5-6), pp. 1091–1101.
  • C. W. Clenshaw, G. F. Miller, and M. Woodger (1962) Algorithms for special functions. I. Numer. Math. 4, pp. 403–419.
  • C. W. Clenshaw and F. W. J. Olver (1984) Beyond floating point. J. Assoc. Comput. Mach. 31 (2), pp. 319–328.
  • C. W. Clenshaw (1955) A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.
  • 5: 28.5 Second Solutions fe n , ge n
    (Other normalizations for C n ( q ) and S n ( q ) can be found in the literature, but most formulas—including connection formulas—are unaffected since fe n ( z , q ) / C n ( q ) and ge n ( z , q ) / S n ( q ) are invariant.) …
    See accompanying text
    Figure 28.5.1: fe 0 ( x , 0.5 ) for 0 x 2 π and (for comparison) ce 0 ( x , 0.5 ) . Magnify
    See accompanying text
    Figure 28.5.2: fe 0 ( x , 1 ) for 0 x 2 π and (for comparison) ce 0 ( x , 1 ) . Magnify
    See accompanying text
    Figure 28.5.3: fe 1 ( x , 0.5 ) for 0 x 2 π and (for comparison) ce 1 ( x , 0.5 ) . Magnify
    See accompanying text
    Figure 28.5.4: fe 1 ( x , 1 ) for 0 x 2 π and (for comparison) ce 1 ( x , 1 ) . Magnify
    6: 18.40 Methods of Computation
    However, 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. For further information see Clenshaw (1955), Gautschi (2004, §§2.1, 8.1), and Mason and Handscomb (2003, §2.4). …
    7: 2.10 Sums and Sequences
    §2.10(i) Euler–Maclaurin Formula
    Another version is the Abel–Plana formula: … We need a “comparison function” g ( z ) with the properties: …
  • (c)

    The coefficients in the Laurent expansion

    2.10.27 g ( z ) = n = - g n z n , 0 < | z | < r ,

    have known asymptotic behavior as n ± .

  • (b´)

    On the circle | z | = r , the function f ( z ) - g ( z ) has a finite number of singularities, and at each singularity z j , say,

    2.10.30 f ( z ) - g ( z ) = O ( ( z - z j ) σ j - 1 ) , z z j ,

    where σ j is a positive constant.

  • 8: Bibliography W
  • J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
  • C. A. Wills, J. M. Blair, and P. L. Ragde (1982) Rational Chebyshev approximations for the Bessel functions J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) . Math. Comp. 39 (160), pp. 617–623.
  • J. Wimp (1968) Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.
  • R. Wong (1982) Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.
  • 9: Bibliography P
  • V. I. Pagurova (1965) An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).
  • R. Piessens and M. Branders (1983) Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.
  • A. Poquérusse and S. Alexiou (1999) Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.
  • J. L. Powell (1947) Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.
  • 10: 29.20 Methods of Computation
    Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … §29.15(i) includes formulas for normalizing the eigenvectors. …