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11: Bibliography L

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Comparison of a pair of upper bounds for a ratio of gamma functions. Math. Balkanica (N.S.) 16 (1-4), pp. 195–202.

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External Links:

ISSN 0205-3217, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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12: 19.9 Inequalities

… ►The lower bound in (19.9.4) is sharper than $2/\pi$ when $0\le k^2\le 0.9960$. … ►(19.9.15) is useful when $k^2$ and ${\sin }^2\varphi$ are both close to $1$, since the bounds are then nearly equal; otherwise (19.9.14) is preferable. … ►

19.9.17 $$L\le F(\varphi ,k)\le \sqrt{UL}\le \frac{1}{2}(U+L)\le U,$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $L$: lower limit and $U$: upper limit

Permalink:

http://dlmf.nist.gov/19.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

►where …

13: 1.4 Calculus of One Variable

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1.4.33 $${𝒱}_{a,b}\left(f\right)=sup\sum _{j=1}^n\left|f(x_j)-f(x_{j-1})\right|,$$

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Defines:

$𝒱\left(f\right)$: total variation and ${𝒱}_{a,b}\left(f\right)$: total variation

Symbols:

$sup$: least upper bound (supremum), $j$: integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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14: 2.3 Integrals of a Real Variable

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2.3.3 $${\sigma }_n=\underset{(0,\mathrm{\infty })}{sup}(t^{-1}\ln |q^{(n)}(t)/q^{(n)}(0)|)$$

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Defines:

${\sigma }_n$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $sup$: least upper bound (supremum), $q(t)$: infinitely differentiable function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/2.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(i), §2.3 and Ch.2

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15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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1.18.22

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Symbols:

$\in$: element of, $\equiv$: equals by definition, $\Vert 𝐀\Vert$: matrix norm, $sup$: least upper bound (supremum) and $\Vert 𝐯\Vert$: vector norm ($l$²)

Permalink:

http://dlmf.nist.gov/1.18.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

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1.18.68

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Symbols:

${𝐀}^{\ast }$: adjoint of matrix, $\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $sup$: least upper bound (supremum), $\Vert 𝐯\Vert$: vector norm ($l$²), $w$: variable and $\left|x\right|$: absolute value of $x$

Notes:

See Dunford and Schwartz (1988, §XII.1.4).

Permalink:

http://dlmf.nist.gov/1.18.E68

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ix), §1.18(ix), §1.18 and Ch.1

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16: DLMF Project News

error generating summary

17: 2.5 Mellin Transform Methods

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2.5.34 $$\underset{p_{jk}\le x\le q_{jk}}{sup}\left|G_{jk}(x+\mathrm{i}y)\right|\to 0,$$ $y\to \pm \mathrm{\infty }$,

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Symbols:

$\mathrm{i}$: imaginary unit, $sup$: least upper bound (supremum), $G_{jk}(z)$: function, $p_{jk}$: real number and $q_{jk}>p_{jk}$: real number

Permalink:

http://dlmf.nist.gov/2.5.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

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18: 19.24 Inequalities

… ►The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. …

19: 19.27 Asymptotic Approximations and Expansions

… ►The approximations in §§19.27(i)–19.27(v) are furnished with upper and lower bounds by Carlson and Gustafson (1994), sometimes with two or three approximations of differing accuracies. …

20: Errata

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The upper and lower bounds given have been replaced with stronger bounds.

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