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11: 3.1 Arithmetics and Error Measures

… ►For given values of $E_{\min }$, $E_{\max }$, and $p$, the format width in bits $N$ of a computer word is the total number of bits: the sign (one bit), the significant bits $b_1,b_2,\mathrm{\dots },b_{p-1}$ ($p-1$ bits), and the bits allocated to the exponent (the remaining $N-p$ bits). … ►Interval arithmetic is intended for bounding the total effect of rounding errors of calculations with machine numbers. …

12: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.11 $$|R_{\mathrm{\ell }}(\nu ,z)|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\mathrm{\infty }}\left(t^{-1}\right)\right),$$

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Defines:

$R_{\mathrm{\ell }}(\nu ,z)$: remainder (locally)

Symbols:

$\exp z$: exponential function, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.40.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

… ►

10.40.12

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, ${𝒱}_{a,b}\left(f\right)$: total variation and $z$: complex variable

Referenced by:

§10.40(i), §10.40(iii), Erratum (V1.0.11) for Equation (10.40.12)

Permalink:

http://dlmf.nist.gov/10.40.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.0.11):

Originally the third constraint was written incorrectly as $\pi \le |\mathrm{ph}z|\le \frac{3}{2}\pi$.

Suggested 2014-11-05 by Gergő Nemes

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

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13: Bibliography R

… ►

Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.

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External Links:

ISSN 0022-4715, Document, MathReview (Stamatis Koumandos), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

…

14: 10.17 Asymptotic Expansions for Large Argument

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10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right),$$

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Defines:

$R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally)

Symbols:

$\exp z$: exponential function, $\mathrm{i}$: imaginary unit, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.17(iv), Erratum (V1.0.10) for Equation (10.17.14)

Permalink:

http://dlmf.nist.gov/10.17.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

… ►

10.17.15

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable and $\chi (\mathrm{\ell })$

Referenced by:

§10.17(iv)

Permalink:

http://dlmf.nist.gov/10.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

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15: 13.9 Zeros

… ►Let $T(a,b)$ be the total number of zeros in the sector , $P(a,b)$ be the corresponding number of positive zeros, and $a$, $b$, and $a-b+1$ be nonintegers. …

16: 26.10 Integer Partitions: Other Restrictions

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26.10.10 $$p(𝒟k,n)=\sum p_m\left(n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\right),$$

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Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(iii), §26.10 and Ch.26

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17: 2.7 Differential Equations

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2.7.23 $$|{ϵ}_j(x)|,\frac{1}{2}{f}^{-1/2}(x)|{ϵ}_j^{\prime }(x)|\le \exp \left(\frac{1}{2}{𝒱}_{a_j,x}\left(F\right)\right)-1,$$ $j=1,2$,

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Symbols:

$\exp z$: exponential function, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval, ${ϵ}_j(x)$: function and $F$: error-control function

Permalink:

http://dlmf.nist.gov/2.7.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

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2.7.25 $${𝒱}_{a_j,x}\left(F\right)=\left|{\int }_{a_j}^x\left|\frac{1}{f^{1/4}(t)}\frac{d^2}{{dt}^2}\left(\frac{1}{f^{1/4}(t)}\right)-\frac{g(t)}{f^{1/2}(t)}\right|dt\right|.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval and $F$: error-control function

Referenced by:

Erratum (V1.1.2) for Equation (2.7.25)

Permalink:

http://dlmf.nist.gov/2.7.E25

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value.

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

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18: Bibliography G

… ►

The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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External Links:

FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

Encodings:

BibTeX

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19: 31.2 Differential Equations

… ►The total number of free parameters is six. …

20: 1.17 Integral and Series Representations of the Dirac Delta

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