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31: 2.1 Definitions and Elementary Properties

… ►(In other words $=$ here really means $\subseteq$.) … ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. …

32: 1.9 Calculus of a Complex Variable

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Mean Value Property

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33: 3.11 Approximation Techniques

… ►where $x_{\mathrm{\ell }}=\cos \left(\pi \mathrm{\ell }/n\right)$ and the double prime means that the first and last terms are to be halved. … ►Here the single prime on the summation symbol means that the first term is to be halved. … ►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. …

34: Bibliography F

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:

FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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35: Bibliography C

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The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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External Links:

ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.8(i).

Encodings:

BibTeX

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Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:

ISSN 0002-9920, MathReview (Willard Parker)

Referenced by:

§19.8(i).

Encodings:

BibTeX

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36: 9.12 Scorer Functions

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9.12.15 $$\mathrm{Gi}\left(z\right)=\frac{3^{-2/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\cos \left(\frac{2k-1}{3}\pi \right)\mathrm{\Gamma }\left(\frac{k+1}{3}\right)\frac{{(3^{1/3}z)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Substitute into (9.12.11) by means of (9.12.17), (9.4.3), (9.2.5), (9.2.6), and use (5.5.3).

Referenced by:

(9.12.16)

Permalink:

http://dlmf.nist.gov/9.12.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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9.12.17 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{k+1}{3}\right)\frac{{(3^{1/3}z)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Expand the integral in (9.12.20) in powers of $zt$ and integrate term-by-term by means of (5.2.1).

Referenced by:

(9.12.15), (9.12.18)

Permalink:

http://dlmf.nist.gov/9.12.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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37: 10.43 Integrals

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38: Bibliography D

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The computation of lattice frequency distribution functions by means of continued fractions. Physica 39 (3), pp. 413–423.

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External Links:

ISSN 0031-8914, Document, Link

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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39: 3.5 Quadrature

… ►If $k$ in (3.5.4) is not arbitrarily large, and if odd-order derivatives of $f$ are known at the end points $a$ and $b$, then the composite trapezoidal rule can be improved by means of the Euler–Maclaurin formula (§2.10(i)). … ►These can be found by means of the recursion … ►When $f\in C^{\mathrm{\infty }}$, the Romberg method affords a means of obtaining high accuracy in many cases with a relatively simple adaptive algorithm. …

40: 10.41 Asymptotic Expansions for Large Order

… ►To establish (10.41.12) we substitute into (10.34.3), with $m=0$ and $z$ replaced by $\nu z$, by means of (10.41.13) observing that when $|z|$ is large the effect of replacing $z$ by $z{\mathrm{e}}^{\pm \pi \mathrm{i}}$ is to replace $\eta$, ${(1+z^2)}^{\frac{1}{4}}$, and $p$ by $-\eta$, $\pm \mathrm{i}{(1+z^2)}^{\frac{1}{4}}$, and $-p$, respectively. …