relation to minimax polynomials

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21: Mourad E. H. Ismail

… ►Ismail has published numerous papers on special functions, orthogonal polynomials, approximation theory, combinatorics, asymptotics, and related topics. His well-known book Classical and Quantum Orthogonal Polynomials in One Variable was published by Cambridge University Press in 2005 and reprinted with corrections in paperback in Ismail (2009). … ► 190, American Mathematical Society, 1995; Special Functions, $q$-Series and Related Topics (with D. … Garvan), Kluwer Academic Publishers, 2001; and Theory and Applications of Special Functions: A volume dedicated to Mizan Rahman (with E. …

22: 7.24 Approximations

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Hastings (1955) gives several minimax polynomial and rational approximations for $\mathrm{erf}x$, $\mathrm{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$.

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

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Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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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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Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi }\mathrm{erf}x$ and ${\mathrm{e}}^{x^2}F\left(x\right)$ for $-3\le x\le 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi }x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ and $2xF\left(x\right)$ for $x\ge 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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23: 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 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 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. … ►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 $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

24: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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Jacobi

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Legendre

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Meixner

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25: 7.10 Derivatives

§7.10 Derivatives

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7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

►For the Hermite polynomial $H_n\left(z\right)$ see §18.3. ►

7.10.2 $$w^{\prime }\left(z\right)=-2zw\left(z\right)+(2\mathrm{i}/\sqrt{\pi }),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.20

Permalink:

http://dlmf.nist.gov/7.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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26: 24.4 Basic Properties

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§24.4(ii) Symmetry

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§24.4(vii) Derivatives

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§24.4(ix) Relations to Other Functions

►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

27: 18.5 Explicit Representations

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Chebyshev

… ►Related formula: … ►

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Laguerre

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Hermite

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28: 29 Lamé Functions

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29: 18.11 Relations to Other Functions

§18.11 Relations to Other Functions

… ►See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions. ►

Ultraspherical

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Laguerre

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Hermite

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30: 18.21 Hahn Class: Interrelations

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§18.21(ii) Limit Relations and Special Cases

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Hahn $\to$ Jacobi

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Meixner $\to$ Laguerre

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Charlier $\to$ Hermite

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Meixner–Pollaczek $\to$ Laguerre

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