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1: 10.37 Inequalities; Monotonicity

… ►If $\nu$ $(\ge 0)$ is fixed, then throughout the interval , $I_{\nu }\left(x\right)$ is positive and increasing, and $K_{\nu }\left(x\right)$ is positive and decreasing. ►If $x$ $(>0)$ is fixed, then throughout the interval , $I_{\nu }\left(x\right)$ is decreasing, and $K_{\nu }\left(x\right)$ is increasing. …

2: 33.23 Methods of Computation

… ►On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►This implies decreasing $\mathrm{\ell }$ for the regular solutions and increasing $\mathrm{\ell }$ for the irregular solutions of §§33.2(iii) and 33.14(iii). …

3: 7.8 Inequalities

… ►The function $F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2x^2}}$ is strictly decreasing for $x>0$. …

4: 8.13 Zeros

… ►The negative zero $x_{-}(a)$ decreases monotonically in the interval , and satisfies …

5: 1.4 Calculus of One Variable

… ►

§1.4(i) Monotonicity

… ►Similarly for nonincreasing and decreasing (strictly decreasing) functions. Each of the preceding four cases is classified as monotonic; sometimes strictly monotonic is used for the strictly increasing or strictly decreasing cases. …

6: 26.12 Plane Partitions

… ►A plane partition, $\pi$, of a positive integer $n$, is a partition of $n$ in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … ►A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns. …

7: 2.6 Distributional Methods

… ►, a continuous linear functional) on the space $𝒯$ of rapidly decreasing functions on $ℝ$. … ►

2.6.11 $$⟨f,\varphi ⟩={\int }_0^{\mathrm{\infty }}f(t)\varphi (t)dt,$$ $\varphi \in 𝒯$.

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Symbols:

$dx$: differential, $⟨\mathrm{\Lambda },\varphi ⟩$: inner-product of distribution, $\in$: element of, $\int$: integral, $f(t)$: locally integrable function and $𝒯$: space of decreasing functions

Referenced by:

§2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.12 $$⟨t^{-\alpha },\varphi ⟩={\int }_0^{\mathrm{\infty }}{t}^{-\alpha }\varphi (t)dt,$$ $\varphi \in 𝒯$,

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Symbols:

$dx$: differential, $⟨\mathrm{\Lambda },\varphi ⟩$: inner-product of distribution, $\in$: element of, $\int$: integral and $𝒯$: space of decreasing functions

Referenced by:

§2.6(iii), §2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.13 $$⟨t^{-s-\alpha },\varphi ⟩=\frac{1}{{\left(\alpha \right)}_s}{\int }_0^{\mathrm{\infty }}{t}^{-\alpha }{\varphi }^{(s)}(t)dt,$$ $\varphi \in 𝒯$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential, $⟨\mathrm{\Lambda },\varphi ⟩$: inner-product of distribution, $\in$: element of, $\int$: integral and $𝒯$: space of decreasing functions

Referenced by:

§2.6(ii), §2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.16 $$⟨f_n,\varphi ⟩={(-1)}^n{\int }_0^{\mathrm{\infty }}{f}_{n,n}(t){\varphi }^{(n)}(t)dt,$$ $\varphi \in 𝒯$,

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Symbols:

$dx$: differential, $⟨\mathrm{\Lambda },\varphi ⟩$: inner-product of distribution, $\in$: element of, $\int$: integral, $n$: nonnegative integer, $f_n(t)$: remainder, $𝒯$: space of decreasing functions and $f_{n,n}(t)$: $n$th repeated integral

Referenced by:

§2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

…

8: 19.30 Lengths of Plane Curves

… ►As $\lambda$ increases, the eccentricity $k$ decreases and the rate of change of arclength for a fixed value of $\varphi$ is given by …

9: 18.16 Zeros

… ►Then ${\theta }_{n,m}$ is strictly increasing in $\alpha$ and strictly decreasing in $\beta$; furthermore, if $\alpha =\beta$, then ${\theta }_{n,m}$ is strictly increasing in $\alpha$. … ►Arrange them in decreasing order: …

10: 4.13 Lambert $W$-Function

… ►On the $z$-interval $(-{\mathrm{e}}^{-1},0)$ there are two real solutions, one increasing and the other decreasing. … ►The decreasing solution can be identified as $W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$. …