expansions in Chebyshev series

SearchAdvancedHelp

(0.005 seconds)

1—10 of 27 matching pages

1: 12.20 Approximations

§12.20 Approximations

…

2: 9.19 Approximations

… ►

§9.19(ii) Expansions in Chebyshev Series

… ►

MacLeod (1994) supplies Chebyshev-series expansions to cover $\mathrm{Gi}\left(x\right)$ for and $\mathrm{Hi}\left(x\right)$ for . The Chebyshev coefficients are given to 20D.

3: 11.15 Approximations

… ►

§11.15(i) Expansions in Chebyshev Series

…

4: 5.23 Approximations

… ►

§5.23(ii) Expansions in Chebyshev Series

…

5: 7.24 Approximations

… ►

§7.24(ii) Expansions in Chebyshev Series

…

6: 6.20 Approximations

… ►

§6.20(ii) Expansions in Chebyshev Series

… ►

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

… ►

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 the scheme can be used in backward direction.

…

7: 25.20 Approximations

… ►

Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta \left(s\right)$ for $0\le s\le 1$ (15D), $\zeta \left(s+1\right)$ for $0\le s\le 1$ (20D), and $\ln \xi \left(\frac{1}{2}+\mathrm{i}x\right)$ (§25.4) for $-1\le x\le 1$ (20D). For errata see Piessens and Branders (1972).

…

8: 3.11 Approximation Techniques

… ►

§3.11(ii) Chebyshev-Series Expansions

… ►Since $L_0=1$, $L_n$ is a monotonically increasing function of $n$, and (for example) $L_{1000}=4.07\mathrm{\dots }$, this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. … ►For further details on Chebyshev-series expansions in the complex plane, see Mason and Handscomb (2003, §5.10). …

9: 7.6 Series Expansions

§7.6 Series Expansions

►

§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

… ►

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

ⓘ

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

…

10: 18.18 Sums

… ►

Chebyshev

… ►

Chebyshev

… ►

Legendre and Chebyshev

…