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1: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …

2: 3.1 Arithmetics and Error Measures

… ►The lower and upper bounds for the absolute values of the nonzero machine numbers are given by … ►Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases. … ►If $x^{\ast }$ is an approximation to a real or complex number $x$, then the absolute error is ►

3.1.8 $${ϵ}_a=\left|x^{\ast }-x\right|.$$

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Defines:

${ϵ}_a$: absolute error (locally)

A&S Ref:

3.5.1 (in absolute value)

Permalink:

http://dlmf.nist.gov/3.1.E8

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Annotations for §3.1(v), §3.1 and Ch.3

… ►

3.1.9 $${ϵ}_r=\left|\frac{x^{\ast }-x}{x}\right|=\frac{{ϵ}_a}{\left|x\right|}.$$

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Defines:

${ϵ}_r$: relative error (locally)

Symbols:

${ϵ}_a$: absolute error

A&S Ref:

3.5.2 (in absolute value)

Permalink:

http://dlmf.nist.gov/3.1.E9

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Annotations for §3.1(v), §3.1 and Ch.3

…

3: 6.13 Zeros

… ► $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ each have an infinite number of positive real zeros, which are denoted by $c_k$, $s_k$, respectively, arranged in ascending order of absolute value for $k=0,1,2,\mathrm{\dots }$. …

4: 23.16 Graphics

… ►In Figures 23.16.2 and 23.16.3, height corresponds to the absolute value of the function and color to the phase. …

5: 1.9 Calculus of a Complex Variable

… ►

1.9.12 $$\left|\overline{z}\right|=\left|z\right|,$$

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Symbols:

$\overline{z}$: complex conjugate, $z$: variable and $\left|x\right|$: absolute value of $x$

A&S Ref:

3.7.8

Permalink:

http://dlmf.nist.gov/1.9.E12

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Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

… ►

§1.9(v) Infinite Sequences and Series

… ►

1.9.47

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Symbols:

$z$: variable, $n$: nonnegative integer, $ϵ$: positive number and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.9.E47

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Annotations for §1.9(v), §1.9 and Ch.1

… ►For $z$ in $\left|z-z_0\right|\le \rho$ (), the convergence is absolute and uniform. … ►

1.9.64

ⓘ

Symbols:

$z$: variable, $m$: nonnegative integer, $n$: nonnegative integer, $ϵ$: positive number and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.9.E64

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Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

…

6: 1.10 Functions of a Complex Variable

… ►

1.10.2 $${\mathrm{e}}^z=1+\frac{z}{1!}+\frac{z^2}{2!}+\mathrm{\cdots },$$ ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: variable and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.10.E2

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Annotations for §1.10(i), §1.10(i), §1.10 and Ch.1

►

1.10.3 $$\ln \left(1+z\right)=z-\frac{z^2}{2}+\frac{z^3}{3}-\mathrm{\cdots },$$ ,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $z$: variable and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.10.E3

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Annotations for §1.10(i), §1.10(i), §1.10 and Ch.1

… ►

§1.10(ix) Infinite Products

… ►The convergence of the product is absolute if ${\prod }_{n=1}^{\mathrm{\infty }}(1+\left|a_n\right|)$ converges. … ►

1.10.26 $$F(x;z)=\sum _{n=0}^{\mathrm{\infty }}{p}_n(x)z^n,$$ .

ⓘ

Symbols:

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

Referenced by:

Erratum (V1.2.0) §1.10

Permalink:

http://dlmf.nist.gov/1.10.E26

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Annotations for §1.10(xi), §1.10 and Ch.1

…

7: 1.4 Calculus of One Variable

… ►

1.4.2

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/1.4.E2

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

… ►Absolute convergence also implies convergence. … ► … ►In particular, absolute continuity occurs if the function $\alpha (x)$ is differentiable, ${\alpha }^{\prime }(x)=w(x)$ with $w(x)$ continuous. …

8: 1.15 Summability Methods

… ►

1.15.28 $$-\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{i}(\mathrm{sign}n)F(n)r^{\left|n\right|}{\mathrm{e}}^{\mathrm{i}n\theta };$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{sign}x$: sign of, $n$: nonnegative integer, $F(n)$ and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E28

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Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

… ►

1.15.30 $$\underset{ϵ\to 0+}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-ϵ\left|t\right|}f(t)dt=L.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E30

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Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

… ►

1.15.32 $$\underset{R\to \mathrm{\infty }}{lim}{\int }_{-R}^R\left(1-\frac{\left|t\right|}{R}\right)f(t)dt=L.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/1.15.E32

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Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

… ►

1.15.35 $${\int }_{\left|x\right|\ge \delta }P(x,y)dx\to 0,$$ as $y\to 0$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $P(x,y)$: Poisson kernel and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E35

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Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

… ►

1.15.38 $$\underset{y\to 0+}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|h(x,y)-f(x)\right|dx=0.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $h(x,y)$: function and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E38

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Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

…

9: 5.3 Graphics

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

10: 8.27 Approximations

… ►

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 $x\to \mathrm{\infty }$.

…