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11: 22.10 Maclaurin Series

… ►The full expansions converge when . …

12: 31.8 Solutions via Quadratures

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31.8.3 $$g=\frac{1}{2}\max (2\underset{0\le k\le 3}{\max }{m}_k,1+N-(1+{(-1)}^N)\left(\frac{1}{2}+\underset{0\le k\le 3}{\min }{m}_k\right)).$$

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Symbols:

$m$: nonnegative integer, $g$: degree of $\lambda$ and $N$: degree of $z$

Permalink:

http://dlmf.nist.gov/31.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.8 and Ch.31

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13: 28.29 Definitions and Basic Properties

… ► $\pi$ is the minimum period of $Q$. … ►If (28.29.1) has a periodic solution with minimum period $n\pi$, $n=3,4,\mathrm{\dots }$, then all solutions are periodic with period $n\pi$. …

14: Bibliography N

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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

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Notes:

Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.

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Notes:

Extended-precision Fortran evaluator, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§13.32(iii).

Encodings:

BibTeX

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15: 2.4 Contour Integrals

… ►Now suppose that in (2.4.10) the minimum of $\mathrm{\Re }\left(zp(t)\right)$ on $𝒫$ occurs at an interior point $t_0$. … ►Cases in which $p^{\prime }(t_0)\ne 0$ are usually handled by deforming the integration path in such a way that the minimum of $\mathrm{\Re }\left(zp(t)\right)$ is attained at a saddle point or at an endpoint. … ►In the commonest case the interior minimum $t_0$ of $\mathrm{\Re }\left(zp(t)\right)$ is a simple zero of $p^{\prime }(t)$. …

16: 1.7 Inequalities

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1.7.8 $$\min (a_1,a_2,\mathrm{\dots },a_n)\le M(r)\le \max (a_1,a_2,\mathrm{\dots },a_n),$$

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Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.2.2

Permalink:

http://dlmf.nist.gov/1.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §1.7(iii), §1.7 and Ch.1

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17: 2.3 Integrals of a Real Variable

… ►derives from the neighborhood of the minimum of $p(t)$ in the integration range. Without loss of generality, we assume that this minimum is at the left endpoint $a$. … ►

$p^{\prime }(t)$ and $q(t)$ are continuous in a neighborhood of $a$, save possibly at $a$, and the minimum of $p(t)$ in $[a,b)$ is approached only at $a$.

… ►Assume also that ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are continuous in $\alpha$ and $t$, and for each $\alpha$ the minimum value of $p(\alpha ,t)$ in $[0,k)$ is at $t=\alpha$, at which point $\partial p(\alpha ,t)/\partial t$ vanishes, but both ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are nonzero. …

18: 6.12 Asymptotic Expansions

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19: 36.8 Convergent Series Expansions

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$a_{n+1}(𝐱)={\displaystyle \frac{\mathrm{i}}{n+1}}{\displaystyle \sum _{p=0}^{\min (n,K-1)}}(p+1)x_{p+1}{a}_{n-p}(𝐱),$ $n=0,1,2,\mathrm{\dots }$.

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20: 34.5 Basic Properties: $\mathit{6}j$ Symbol

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34.5.19 $2\mu -j$ odd, $\mu =\min (j_1,j_2)$,

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Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.5.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(vi), §34.5 and Ch.34

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34.5.20 $\mu =\min (j_1,j_2)$,

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Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(vi), §34.5 and Ch.34

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