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31: 2.3 Integrals of a Real Variable

… ►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$. … ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by ►

2.3.6 $${𝒱}_{a,b}\left(f(t)\right)={\int }_a^b\left|f^{\prime }(t)\right|dt;$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $a$: left endpoint and $b$: right endpoint

Referenced by:

§10.17(iv), Erratum (V1.1.1) for Equation (2.3.6)

Permalink:

http://dlmf.nist.gov/2.3.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.1):

The integrand was corrected so that the absolute value does not include the differential.

Suggested 2021-02-08 by Juan Luis Varona

See also:

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

…

32: 18.15 Asymptotic Approximations

… ►When $\alpha ,\beta \in (-\frac{1}{2},\frac{1}{2})$, the error term in (18.15.1) is less than twice the first neglected term in absolute value, in which one has to take $\cos {\theta }_{n,m,\mathrm{\ell }}=1$. …

33: 9.9 Zeros

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►They lie in the sectors and , and are denoted by ${\beta }_k$, ${\beta }_k^{\prime }$, respectively, in the former sector, and by $\overline{{\beta }_k}$, $\overline{{\beta }_k^{\prime }}$, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\mathrm{\dots }.$ See §9.3(ii) for visualizations. …

34: 10.17 Asymptotic Expansions for Large Argument

… ►Then the remainder associated with the sum ${\sum }_{k=0}^{\mathrm{\ell }-1}{(-1)}^k{a}_{2k}(\nu )z^{-2k}$ does not exceed the first neglected term in absolute value and has the same sign provided that $\mathrm{\ell }\ge \max (\frac{1}{2}\nu -\frac{1}{4},1)$. … ►If these expansions are terminated when $k=\mathrm{\ell }-1$, then the remainder term is bounded in absolute value by the first neglected term, provided that $\mathrm{\ell }\ge \max (\nu -\frac{1}{2},1)$. …

35: Mathematical Introduction

… ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …

36: 1.16 Distributions

… ►

1.16.24 $$\left|x^N{\varphi }_n^{(k)}\right|\le c_{k,N}$$

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Symbols:

$k$: integer, $n$: nonnegative integer, $\varphi (x)$: test function and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.16.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(v), §1.16 and Ch.1

… ►

1.16.27 $$⟨{𝐷}^{k}f,\varphi ⟩={(-1)}^{\left|k\right|}{\int }_{{ℝ}^n}f(x){\varphi }^{(k)}(x)dx,$$ $\varphi \in {𝒟}_n$,

ⓘ

Symbols:

$dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral, $ℝ$: real line, $k$: integer, $n$: nonnegative integer, ${𝒟}_n$: space of test functions, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.16.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(vi), §1.16 and Ch.1

… ►

1.16.28 $$\left|x^m{\varphi }^{(k)}(x)\right|\le c_{m,k},$$ $x\in {ℝ}^n$.

ⓘ

Symbols:

$\in$: element of, $ℝ$: real line, $k$: integer, $m$: nonnegative integer, $n$: nonnegative integer, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.16.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(vi), §1.16 and Ch.1

… ►

1.16.30 $$𝐃=(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1},\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_2},\mathrm{\dots },\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}).$$

ⓘ

Defines:

$𝐃$: vector differential operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Referenced by:

(1.16.32), item (ii), item (ii)

Permalink:

http://dlmf.nist.gov/1.16.E30

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was completely different:

$${𝐷}_{𝜶}={\mathrm{i}}^{-\left|𝜶\right|}{𝐷}^{𝜶}={\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\cdots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

The change was made to clarify the meaning of (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

…

37: 8.3 Graphics

… ►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …

38: 10.3 Graphics

… ►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …

39: 23.4 Graphics

… ►Height corresponds to the absolute value of the function and color to the phase. …

40: 23.22 Methods of Computation

… ►For $2{\omega }_1$ choose a nonzero point of $𝕃$ of smallest absolute value. …