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11: 8.27 Approximations

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DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

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12: 9.3 Graphics

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

13: 14.22 Graphics

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

14: 7.13 Zeros

… ► $\mathrm{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $ℂ$ there is an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►In the sector , $\mathrm{erfc}z$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►In the first quadrant of $ℂ$ $C\left(z\right)$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. …

15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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1.18.2 $$\sum _n{\left|c_n\right|}^2\le {\Vert v\Vert }^2,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/1.18.E2

Encodings:

pMML, png, TeX

See also:

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

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1.18.4 $$\sum _n{\left|c_n\right|}^2={\Vert v\Vert }^2,$$

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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:

§1.18(i)

Permalink:

http://dlmf.nist.gov/1.18.E4

Encodings:

pMML, png, TeX

See also:

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

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1.18.5

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Symbols:

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

Referenced by:

§1.18(i)

Permalink:

http://dlmf.nist.gov/1.18.E5

Encodings:

pMML, png, TeX

See also:

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

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1.18.8

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Symbols:

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

Permalink:

http://dlmf.nist.gov/1.18.E8

Encodings:

pMML, png, TeX

See also:

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

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1.18.11

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E11

Encodings:

pMML, png, TeX

See also:

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

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16: 1.2 Elementary Algebra

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1.2.45 $${\Vert 𝐯\Vert }_p={\left(\sum _{i=1}^n{\left|v_i\right|}^p\right)}^{1/p},$$ $p\ge 1$.

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Defines:

${\Vert 𝐯\Vert }_p$: L-p norm

Symbols:

$n$: nonnegative integer, $𝐯$: column vector and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.2.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.47 $${\Vert 𝐯\Vert }_1=\sum _{i=1}^n\left|v_i\right|,$$

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Defines:

${\Vert 𝐯\Vert }_1$: l1 norm

Symbols:

$n$: nonnegative integer, $𝐯$: column vector and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.2.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.48 $${\Vert 𝐯\Vert }_{\mathrm{\infty }}=\max (\left|v_1\right|,\left|v_2\right|,\mathrm{\dots },\left|v_n\right|).$$

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Defines:

${\Vert 𝐯\Vert }_{\mathrm{\infty }}$: l-infinity norm

Symbols:

$n$: nonnegative integer, $𝐯$: column vector and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.2.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.51 $$\left|⟨𝐮,𝐯⟩\right|\le \Vert 𝐮\Vert \Vert 𝐯\Vert ,$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $\Vert 𝐯\Vert$: vector norm ($l$²), $𝐯$: column vector, $𝐮$: column vector and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.2.E51

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.57 $$a_{ij}=0,$$ for $\left|i-j\right|>1$.

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Symbols:

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

Permalink:

http://dlmf.nist.gov/1.2.E57

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

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17: 1.8 Fourier Series

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1.8.5 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\frac{1}{2}{\left|a_0\right|}^2+\sum _{n=1}^{\mathrm{\infty }}({\left|a_n\right|}^2+{\left|b_n\right|}^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E5

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.6 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left|c_n\right|}^2,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.8(i), Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.8 $$L_n=\frac{1}{\pi }{\int }_0^{\pi }\frac{\left|\sin \left(n+\frac{1}{2}\right)t\right|}{\sin \left(\frac{1}{2}t\right)}dt,$$ $n=0,1,\mathrm{\dots }$.

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Defines:

$L_n$: Lebesgue constants (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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18: 12.3 Graphics

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

19: 15.3 Graphics

… ►In Figures 15.3.5 and 15.3.6, height corresponds to the absolute value of the function and color to the phase. …

20: 18.40 Methods of Computation

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Figure 18.40.2: Derivative Rule inversions for $w^{\mathrm{RCP}}(x)$ carried out via Lagrange and PWCF interpolations. Shown are the absolute errors of approximation (18.40.8) at the points $x_{i,N}$, $i=1,2,\mathrm{\dots },N$ for $N=40$. … Magnify

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