expansions in Chebyshev series

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11: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for

K\left(k\right)

and

E\left(k\right)

in terms of

m=k^{2}

with

0\leq m<1

can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for

K\left(k\right)

and

E\left(k\right)

for

0<k\leq 1

are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. ►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for

\phi

near

\pi/2

with the improvements made in the 1970 reference. …

12: 18.38 Mathematical Applications

… ►In consequence, expansions of functions that are infinitely differentiable on

[-1,1]

in series of Chebyshev polynomials usually converge extremely rapidly. …

13: 18.40 Methods of Computation

… ►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. …

14: 8.27 Approximations

… ►

•

Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\Gamma\left(a,\omega z\right)$ (by specifying parameters) with $1\leq\omega<\infty$ , and $\gamma\left(a,\omega z\right)$ with $0\leq\omega\leq 1$ ; see also Temme (1994b, §3).

… ►

•

Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$ .

… ►

•

Verbeeck (1970) gives polynomial and rational approximations for $E_{p}\left(x\right)=(e^{-x}/x)P(z)$ , approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$ .

15: 13.31 Approximations

… ►

§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of

M\left(a,b,x\right)

and

U\left(a,b,x\right)

that include the intervals

0\leq x\leq\alpha

and

\alpha\leq x<\infty

, respectively, where

\alpha

is an arbitrary positive constant. … ►In Luke (1977a) the following rational approximation is given, together with its rate of convergence. … ►

13.31.1

A_{n}(z)=\sum_{s=0}^{n}\frac{{\left(-n\right)_{s}}{\left(n+1\right)_{s}}{\left% (a\right)_{s}}{\left(b\right)_{s}}}{{\left(a+1\right)_{s}}{\left(b+1\right)_{s% }}(n!)^{2}}\*{{}_{3}F_{3}}\left({-n+s,n+1+s,1\atop 1+s,a+1+s,b+1+s};-z\right),

ⓘ

Defines:: $A_{n}(z)$ : expansion (locally)
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.31.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.31(iii), §13.31 and Ch.13

… ►

13.31.3

z^{a}U\left(a,1+a-b,z\right)=\lim_{n\to\infty}\frac{A_{n}(z)}{B_{n}(z)}.

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $n$ : nonnegative integer, $z$ : complex variable, $A_{n}(z)$ : expansion and $B_{n}(z)$ : expansion
Permalink:: http://dlmf.nist.gov/13.31.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §13.31(iii), §13.31 and Ch.13

16: Bibliography S

… ►

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

►

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

… ►

M. J. Seaton (2002b) FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:: Single-precision Fortran code.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 10th item.
Encodings:: BibTeX

… ►

H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §12.13(i).
Encodings:: BibTeX

►

H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

…

17: 18.3 Definitions

§18.3 Definitions

… ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for

n=0,1,\ldots,6

are given in §18.5(iv). … ►

Chebyshev

… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials

T_{n}\left(x\right)

n=0,1,\dots,N

, are orthogonal on the discrete point set comprising the zeros

x_{N+1,n},n=1,2,\dots,N+1

, of

T_{N+1}\left(x\right)

: …

18: Bibliography O

… ►

A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

… ►

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

… ►

F. W. J. Olver (1964b) Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.

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External Links:: Document, MathReview (N. D. Kazarinoff), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii), §2.7(ii).
Encodings:: BibTeX

►

F. W. J. Olver (1994b) The Generalized Exponential Integral. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 497–510.

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External Links:: MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §2.11(iii), §8.19(viii).
Encodings:: BibTeX

…

19: Bibliography R

… ►

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

►

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

… ►

W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.

ⓘ

Notes:: McGraw-Hill Series in Higher Mathematics
External Links:: MathReview (F. Smithies), ZentralBlatt
Referenced by:: §1.18(x), §2.6(i).
Encodings:: BibTeX

►

W. Rudin (1976) Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.

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Notes:: International Series in Pure and Applied Mathematics
External Links:: MathReview, ZentralBlatt
Referenced by:: §1.4(iii).
Encodings:: BibTeX

… ►

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.

ⓘ

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

…

20: Bibliography D

… ►

G. Delic (1979a) Chebyshev expansion of the associated Legendre polynomial

P^{M}_{L}(x)

. Comput. Phys. Comm. 18 (1), pp. 63–71.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 25D
External Links:: Document, Code
Referenced by:: 3rd item.
Encodings:: BibTeX

►

G. Delic (1979b) 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

… ►

P. G. L. Dirichlet (1849) Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:: Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.
External Links:: ISBN 0-8284-0225-6, Link, MathReview
Referenced by:: §27.11.
Encodings:: BibTeX

… ►

T. M. Dunster, R. B. Paris, and S. Cang (1998) On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function. Methods Appl. Anal. 5 (3), pp. 223–247.

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External Links:: ISSN 1073-2772, Pdf, MathReview (John G. Stalker), ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

… ►

T. M. Dunster (1997) Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

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External Links:: ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §2.11(iii), §8.20(ii).
Encodings:: BibTeX

…