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21: 20.3 Graphics

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

22: 1.14 Integral Transforms

… ►

1.14.2 $$\left|F(x)\right|\le \frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|f(t)\right|dt.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $F(x)$: Fourier transform of $f(t)$ and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §1.14(i), §1.14 and Ch.1

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1.14.8 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|F(x)\right|}^{2}dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|f(t)\right|}^{2}dt.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $F(x)$: Fourier transform of $f(t)$ and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.14(i)

Permalink:

http://dlmf.nist.gov/1.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

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1.14.18 $$\left|f(t)\right|\le M{\mathrm{e}}^{\alpha t},$$ .

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Symbols:

$\mathrm{e}$: base of natural logarithm, $M$: constant, $\alpha$: constant and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.14(iii), §1.14(iii), §1.14(iii), §1.4(v)

Permalink:

http://dlmf.nist.gov/1.14.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

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1.14.38 $${\int }_0^{\mathrm{\infty }}{(f(x))}^{2}dx=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|ℳf\left(\frac{1}{2}+\mathrm{i}t\right)\right|}^{2}dt.$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\left|x\right|$: absolute value of $x$

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E38

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

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1.14.45 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|\mathscr{H}f\left(x\right)\right|}^{p}dx\le A_p{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|f(t)\right|}^{p}dt,$$

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Symbols:

$\mathscr{H}\left(f\right)\left(x\right)$: Hilbert transform, $dx$: differential of $x$ , $\int$: integral, $A_p$: coefficient and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.14.E45

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Hilbert transform was changed to $\mathscr{H}f\left(x\right)$ from $\mathscr{H}(f;x)$.

See also:

Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

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23: 1.5 Calculus of Two or More Variables

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1.5.2

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Symbols:

$\left|x\right|$: absolute value of $x$

Referenced by:

§1.9(ii)

Permalink:

http://dlmf.nist.gov/1.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

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1.5.23

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $ϵ$: positive number and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.5.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(iv), §1.5(iv), §1.5 and Ch.1

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1.5.42 $${\iint }_{D}f(x,y)dxdy={\iint }_{D^{\ast }}f(x(u,v),y(u,v))\left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv,$$

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Symbols:

$dx$: differential of $x$ , $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.5.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(vi), §1.5(vi), §1.5 and Ch.1

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1.5.43 $${\iiint }_{D}f(x,y,z)dxdydz={\iiint }_{D^{\ast }}f(x(u,v,w),y(u,v,w),z(u,v,w))\left|\frac{\partial (x,y,z)}{\partial (u,v,w)}\right|dudvdw.$$

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Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $w$: variable and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.5.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(vi), §1.5(vi), §1.5 and Ch.1

…

24: 4.15 Graphics

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

25: 10.74 Methods of Computation

… ►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. …

26: 35.1 Special Notation

… ► ►►►

$a,b$	complex variables.
…
$\left|𝐗\right|$	determinant of $𝐗$ (except when $m=1$ where it means either determinant or absolute value, depending on the context).
…

…

27: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.19 $$\sum _{j,k=-\mathrm{\infty }}^{\mathrm{\infty }}\left|a_{j,k}-{\delta }_{j,k}\right|$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $j$: integer, $k$: integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.3.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(iii), §1.3 and Ch.1

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1.3.21 $${\Vert 𝐮\Vert }^2=\sum _{i=1}^n{\left|c_i\right|}^2,$$

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Symbols:

$\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

Erratum (V1.2.0) Section 1.3

Permalink:

http://dlmf.nist.gov/1.3.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(iv), §1.3(iv), §1.3 and Ch.1

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28: 1.12 Continued Fractions

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1.12.25 $$\left|b_n\right|\ge \left|a_n\right|+1,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.12.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

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1.12.28 $$\sum _{n=1}^{\mathrm{\infty }}\left|b_n\right|=\mathrm{\infty }.$$

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Symbols:

$n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.12.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

…

29: 3.2 Linear Algebra

… ►To avoid instability the rows are interchanged at each elimination step in such a way that the absolute value of the element that is used as a divisor, the pivot element, is not less than that of the other available elements in its column. … ►where $\rho (𝐀{𝐀}^{\mathrm{T}})$ is the largest of the absolute values of the eigenvalues of the matrix $𝐀{𝐀}^{\mathrm{T}}$; see §3.2(iv). …

30: 1.6 Vectors and Vector-Valued Functions

… ►

1.6.50 $$\Vert {𝐓}_{\theta }\times {𝐓}_{\varphi }\Vert ={\rho }^2\left|\sin \theta \right|.$$

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Symbols:

$\sin z$: sine function, $\Vert 𝐯\Vert$: vector norm ($l$²), $\rho$: radius, $\theta$: angle, $\varphi$: angle and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.6.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

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1.6.52 $$A(S)=2\pi {\int }_a^b\left|f(x)\right|\sqrt{1+{(f^{\prime }(x))}^2}dx,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.6.E52

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

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1.6.53 $$A(S)=2\pi {\int }_a^b\left|x\right|\sqrt{1+{(f^{\prime }(x))}^2}dx.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.6.E53

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

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