DLMF: Untitled Document

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Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$ , $5\leq s\leq 11$ , $11\leq s\leq 25$ , $25\leq s\leq 55$ . Precision is varied, with a maximum of 20S.

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J. Oliver (1977) 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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External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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§7.24(ii) Expansions in Chebyshev Series

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Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ for $x\geq 3$ (15D).

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Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x)e^{x^{2}}\operatorname{erfc}x$ on $(0,\infty)$ (22D).

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R. Piessens (1982) 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.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

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R. Piessens (1984a) Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.

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External Links:: ISSN 0021-9991, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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R. Piessens and M. Branders (1972) Chebyshev polynomial expansions of the Riemann zeta function. Math. Comp. 26 (120), pp. G1–G5.

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External Links:: ISSN 0025-5718, FullText, MathReview
Referenced by:: 2nd item, 3rd item.
Encodings:: BibTeX

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A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt
Referenced by:: §1.14(vii).
Encodings:: BibTeX

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P. J. Prince (1975) Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.

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External Links:: ISSN 0098-3500, Document, GAMS (module 498; package TOMS), ZentralBlatt
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

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§6.20(ii) Expansions in Chebyshev Series

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Luke and Wimp (1963) covers $\operatorname{Ei}\left(x\right)$ for $x\leq-4$ (20D), and $\operatorname{Si}\left(x\right)$ and $\operatorname{Ci}\left(x\right)$ for $x\geq 4$ (20D).

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Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\operatorname{Ein}\left(ax\right)$ , $\operatorname{Si}\left(ax\right)$ , and $\operatorname{Cin}\left(ax\right)$ for $-1\leq x\leq 1$ , $a\in\mathbb{C}$ . The coefficients are given in terms of series of Bessel functions.

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Luke (1969b, pp. 321–322) covers $\operatorname{Ein}\left(x\right)$ and $-\operatorname{Ein}\left(-x\right)$ for $0\leq x\leq 8$ (the Chebyshev coefficients are given to 20D); $E_{1}\left(x\right)$ for $x\geq 5$ (20D), and $\operatorname{Ei}\left(x\right)$ for $x\geq 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$ -function (§13.2(i)) from which Chebyshev expansions near infinity for $E_{1}\left(z\right)$ , $\mathrm{f}\left(z\right)$ , and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.

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A. Ralston (1965) Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.

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External Links:: ISSN 0029-599X, Document, MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: §3.11(i).
Encodings:: BibTeX

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M. Razaz and J. L. Schonfelder (1980) High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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Referenced by:: 3rd item.
Encodings:: BibTeX

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M. Razaz and J. L. Schonfelder (1981) Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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External Links:: ISSN 0098-3500, Document
Referenced by:: 1st item.
Encodings:: BibTeX

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E. Ya. Remez (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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Notes:: English translation, AEC-TR-4491, United States Atomic Energy Commission
External Links:: MathReview (A. S. Householder)
Referenced by:: §3.11(iii).
Encodings:: BibTeX

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J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

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A. J. MacLeod (1993) Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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J. C. Mason and D. C. Handscomb (2003) Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton, FL.

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External Links:: ISBN 0-8493-0355-9, MathReview, ZentralBlatt
Referenced by:: §18.1(iii), §18.3, §18.3, Table 18.3.1, §18.38(i), §18.38(i), §18.40(i), (18.5.3), (18.5.4), Table 18.5.1, Table 18.6.1, (18.7.17), (18.7.18), (18.7.5), (18.7.6), (18.9.11), (18.9.12), §3.11(ii), §3.11(ii), §3.11(ii), §3.3(vi), §3.5(iv), Erratum (V1.0.28) for Table 18.3.1.
Encodings:: BibTeX

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J. C. Mason (1993) Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. In Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), Vol. 49, pp. 169–178.

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External Links:: ISSN 0377-0427, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Erratum (V1.0.28) for Table 18.3.1.
Encodings:: BibTeX

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G. F. Miller (1966) On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

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External Links:: ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

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C. W. Clenshaw (1955) A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §18.40(i).
Encodings:: BibTeX

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C. W. Clenshaw (1957) The numerical solution of linear differential equations in Chebyshev series. Proc. Cambridge Philos. Soc. 53 (1), pp. 134–149.

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External Links:: MathReview (L. Fox), ZentralBlatt
Referenced by:: §18.38(i), §3.11(ii).
Encodings:: BibTeX

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C. W. Clenshaw (1962) Chebyshev Series for Mathematical Functions. National Physical Laboratory Mathematical Tables, Vol. 5. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:: MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §3.11(ii), §4.47(i), §5.23(ii), 1st item.
Encodings:: BibTeX

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W. J. Cody (1965b) Chebyshev polynomial expansions of complete elliptic integrals. Math. Comp. 19 (90), pp. 249–259.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.38.
Encodings:: BibTeX

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W. J. Cody (1968) Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.

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Notes:: with Microfiche Supplement
External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

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R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

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External Links:: FullText, MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: §8.7.
Encodings:: BibTeX

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W. G. Bickley and J. Nayler (1935) A short table of the functions

\mathrm{Ki}_{n}(x)

, from

n=1

to

n=16

. Phil. Mag. Series 7 20, pp. 343–347.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.43(iii), 2nd item.
Encodings:: BibTeX

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W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

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J. M. Blair, C. A. Edwards, and J. H. Johnson (1976) Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §7.17(iii), §7.17(iii).
Encodings:: BibTeX

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L. Schoenfeld (1976) Sharper bounds for the Chebyshev functions

\theta(x)

and

\psi(x)

. II. Math. Comp. 30 (134), pp. 337–360.

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Notes:: For corrigendum see same journal v. 30 (1976), no. 136, p. 900
External Links:: FullText, Document, MathReview (R. P. Brent), ZentralBlatt
Referenced by:: §27.12.
Encodings:: BibTeX

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J. L. Schonfelder (1978) Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

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External Links:: FullText, Document, MathReview (L. Fox), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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J. L. Schonfelder (1980) Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.

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External Links:: ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt
Referenced by:: §4.47(i).
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:: Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.
External Links:: ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt
Referenced by:: §3.3(vi), §3.4(i), §3.5(i).
Encodings:: BibTeX

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Chebyshev%20series

1—10 of 12 matching pages

1: 25.20 Approximations

2: Bibliography O

3: 7.24 Approximations

§7.24(ii) Expansions in Chebyshev Series

4: Bibliography P

5: 6.20 Approximations

§6.20(ii) Expansions in Chebyshev Series

6: Bibliography R

7: Bibliography M

8: Bibliography C

9: Bibliography B

10: Bibliography S