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21: Bibliography T

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Parabolic cylinder functions $U(n,x)$ for natural $n$ and positive $x$ . Comput. Phys. Commun. 69, pp. 415–419.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 21D.

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.

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Notes:

Double-precision Fortran, minimum accuracy: 14D. See also Thompson (2004)

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item, §33.23(vi), 3rd item.

Encodings:

BibTeX

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22: 1.5 Calculus of Two or More Variables

… ► $f(x,y)$ has a local minimum (maximum) at $(a,b)$ if …

23: 13.8 Asymptotic Approximations for Large Parameters

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13.8.17 $$M(a,b,z)={\mathrm{e}}^{\nu z}\frac{{\mathrm{\Gamma }}^{\ast }(b)}{{\mathrm{\Gamma }}^{\ast }(a)}\left(1+\frac{(1-\nu )(1+6{\nu }^2{z}^2)}{12a}+O\left(\frac{1}{\min (a^2,b^2)}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(iv), §13.8 and Ch.13

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13.8.18 $$U(a,b+1,z)=z^{-b}{\mathrm{e}}^{(1-\nu )z}\frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(a\right)}\left(1+\frac{\nu z(1-\nu )(2-\nu z)}{2a}+O\left(\frac{1}{\min (a^2,b^2)}\right)\right),$$ $\mathrm{\Re }z>0$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(iv), §13.8 and Ch.13

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24: Bibliography H

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Algorithm 442: Normal deviate. Comm. ACM 16 (1), pp. 51–52.

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Notes:

Includes Algol program, minimum accuracy 24D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(ii).

Encodings:

BibTeX

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Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions: $I_1(x)/I_0(x)$, $I_{1.5}(x)/I_{0.5}(x)$ and their inverses [S14]. ACM Trans. Math. Software 7 (2), pp. 233–238.

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Notes:

Single-precision Fortran, minimum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 571; package TOMS)

Referenced by:

5th item.

Encodings:

BibTeX

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25: 7.12 Asymptotic Expansions

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26: 19.11 Addition Theorems

… ►In the case of $\theta ,\varphi \in [0,\pi /2)$ and , we can use …

27: 19.24 Inequalities

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19.24.12 $$\frac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z})\le R_G(x,y,z)\le \min (\sqrt{\frac{x+y+z}{3}},\frac{x^2+y^2+z^2}{3\sqrt{xyz}}).$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind

Permalink:

http://dlmf.nist.gov/19.24.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19

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28: 9.7 Asymptotic Expansions

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9.7.17

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable, $\zeta$: change of variable, $\chi (x)$: function, $n$: index and $\sigma$: real variable

Proof sketch:

Derivable from (9.6.1)–(9.6.5) and Nemes (2017b, (26) and Propositions B.1, B.3, B.4).

Referenced by:

§9.7(iv), Erratum (V1.0.11) for Equation (9.7.17)

Permalink:

http://dlmf.nist.gov/9.7.E17

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

The bounds have been sharpened for $|\mathrm{ph}z|\le \frac{1}{3}\pi$, from $2\exp \left({\displaystyle \frac{\sigma }{36|\zeta |}}\right)$ to $1$; for $\frac{1}{3}\pi \le |\mathrm{ph}z|\le \frac{2}{3}\pi$, from $2\chi (n)\exp \left({\displaystyle \frac{\sigma \pi }{72|\zeta |}}\right)$ to $\min (|\csc \left(\mathrm{ph}\zeta \right)|,\chi (n+\sigma )+1)$; and for from ${\displaystyle \frac{4\chi (n)}{{|\cos \left(\mathrm{ph}\zeta \right)|}^n}}\exp \left({\displaystyle \frac{\sigma \pi }{36|\mathrm{\Re }\zeta |}}\right)$ to $\frac{\sqrt{2\pi (n+\sigma )}}{{|\cos \left(\mathrm{ph}\zeta \right)|}^{n+\sigma }}+\chi (n+\sigma )+1$.

Errata (effective with 1.0.11):

Originally the constraint condition was written incorrectly as $\frac{2}{3}\pi \le |\mathrm{ph}z|\le \pi$. Also, the equation was reformatted to display the constraints in the equation instead of in the text.

Reported 2014-11-05 by Gergő Nemes

See also:

Annotations for §9.7(iv), §9.7 and Ch.9

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29: 3.6 Linear Difference Equations

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3.6.9 $$\left|\frac{e_N}{p_N{p}_{N+1}}\right|\le ϵ\underset{1\le n\le M}{\min }\left|\frac{e_n}{p_n{p}_{n+1}}\right|.$$

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Symbols:

$N$: arbitrary positive integer, $ϵ$: relative accuracy, $M$: order, $p_n$: solutions and $e_n$: sequence

Referenced by:

§3.6(vi)

Permalink:

http://dlmf.nist.gov/3.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(v), §3.6 and Ch.3

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30: 5.11 Asymptotic Expansions

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5.11.11 $$\left|R_K(z)\right|\le \frac{(1+\zeta \left(K\right))\mathrm{\Gamma }\left(K\right)}{2{(2\pi )}^{K+1}{\left|z\right|}^K}\left(1+\min (\sec \left(\mathrm{ph}z\right),2K^{\frac{1}{2}})\right),$$ $\left|\mathrm{ph}z\right|\le \frac{1}{2}\pi$,

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Defines:

$R_K(z)$: remainder (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\sec z$: secant function and $z$: complex variable

Referenced by:

§5.11(ii)

Permalink:

http://dlmf.nist.gov/5.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(ii), §5.11 and Ch.5

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