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1: 16.26 Approximations

… ►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

2: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►Such a solution is given in terms of a Riemann theta function with two phases. …

3: 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. … ►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. …

4: 33.19 Power-Series Expansions in $r$

… ►

33.19.1 $$f(ϵ,\mathrm{\ell };r)=r^{\mathrm{\ell }+1}\sum _{k=0}^{\mathrm{\infty }}{\alpha }_k{r}^k,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and ${\alpha }_k$: term

Referenced by:

§33.18, §33.19

Permalink:

http://dlmf.nist.gov/33.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

… ►Here $\kappa$ is defined by (33.14.6), $A(ϵ,\mathrm{\ell })$ is defined by (33.14.11) or (33.14.12), ${\gamma }_0=1$, ${\gamma }_1=1$, and … ►

33.19.6 $$k(k+2\mathrm{\ell }+1){\delta }_k+2{\delta }_{k-1}+ϵ{\delta }_{k-2}+2(2k+2\mathrm{\ell }+1)A(ϵ,\mathrm{\ell }){\alpha }_k=0,$$ $k=2,3,\mathrm{\dots }$,

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Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, ${\alpha }_k$: term and ${\delta }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

►with ${\beta }_0={\beta }_1=0$, and ►

33.19.7 $${\beta }_k-{\beta }_{k-1}+\frac{1}{4}(k-1)(k-2\mathrm{\ell }-2)ϵ{\beta }_{k-2}+\frac{1}{2}(k-1)ϵ{\gamma }_{k-2}=0,$$ $k=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, ${\gamma }_k$: term and ${\beta }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

…

5: Guide to Searching the DLMF

… ►

Terms, Phrases and Expressions

►Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: ►

a textual word, a number, or a math symbol.

… ►If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. … ►

Single-letter terms

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6: 12.16 Mathematical Applications

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7: 18.24 Hahn Class: Asymptotic Approximations

… ►In particular, asymptotic formulas in terms of elementary functions are given when $z=x$ is real and fixed. … ►This expansion is in terms of the parabolic cylinder function and its derivative. … ►This expansion is in terms of confluent hypergeometric functions. … ►Both expansions are in terms of parabolic cylinder functions. … ►

Approximations in Terms of Laguerre Polynomials

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8: 6.12 Asymptotic Expansions

… ►For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$. ►For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv), with $p=1$. …If the expansion is terminated at the $n$th term, then the remainder term is bounded by $1+\chi (n+1)$ times the next term. … ►The remainder terms are given by …When $|\mathrm{ph}z|\le \frac{1}{4}\pi$, these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$. …

9: 7.16 Generalized Error Functions

… ►These functions can be expressed in terms of the incomplete gamma function $\gamma (a,z)$ (§8.2(i)) by change of integration variable.

10: 16.7 Relations to Other Functions

… ►Further representations of special functions in terms of ${}_p{}^{}F_q^{}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${}_{q+1}{}^{}F_q^{}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).