Chebyshev-series expansions

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11: 3.11 Approximation Techniques

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§3.11(ii) Chebyshev-Series Expansions

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Chebyshev Expansions

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Calculation of Chebyshev Coefficients

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Complex Variables

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12: 7.24 Approximations

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§7.24(ii) Expansions in Chebyshev Series

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13: 7.6 Series Expansions

§7.6 Series Expansions

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§7.6(i) Power Series

… ►The series in this subsection and in §7.6(ii) converge for all finite values of $|z|$. ►

§7.6(ii) Expansions in Series of Spherical Bessel Functions

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7.6.9 $$\mathrm{erf}\left(az\right)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{(\frac{1}{2}-a^2)z^2}\sum _{n=0}^{\mathrm{\infty }}{T}_{2n+1}\left(a\right){𝗂}_n^{(1)}\left(\frac{1}{2}{z}^2\right),$$ $-1\le a\le 1$.

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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14: 18.3 Definitions

§18.3 Definitions

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15: Bibliography R

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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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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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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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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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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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16: Bibliography D

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Chebyshev expansion of the associated Legendre polynomial $P_L^M(x)$ . Comput. Phys. Comm. 18 (1), pp. 63–71.

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Notes:

Double-precision Fortran, maximum accuracy 25D

External Links:

Document, Code

Referenced by:

3rd item.

Encodings:

BibTeX

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Chebyshev series for the spherical Bessel function $j_l(r)$ . Comput. Phys. Comm. 18 (1), pp. 73–86.

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External Links:

Document

Referenced by:

§10.76(iii).

Encodings:

BibTeX

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The Taylor Series. Oxford University Press, Oxford.

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Notes:

Reprinted by Dover Publications Inc., New York, 1957

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.9(iii).

Encodings:

BibTeX

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Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London-New York.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(11.11.11), §11.6(i), §11.6(iii), §2.11(v), Ch.2, §8.1, Notations, Notations.

Encodings:

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

BibTeX

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17: Bibliography P

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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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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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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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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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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18: Bibliography C

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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:

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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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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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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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Chebyshev expansions for the Bessel function $J_n(z)$ in the complex plane. Math. Comp. 40 (161), pp. 343–366.

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External Links:

ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

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