# absolute

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## 1—10 of 58 matching pages

##### 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 $0\leq m<1$ 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 $0 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 $x^{*}$ is an approximation to a real or complex number $x$, then the absolute error is
3.1.8 $\epsilon_{a}=\left|x^{*}-x\right|.$
3.1.9 $\epsilon_{r}=\left|\frac{x^{*}-x}{x}\right|=\frac{\epsilon_{a}}{\left|x\right|}.$
##### 3: 6.13 Zeros
$\operatorname{Ci}\left(x\right)$ and $\operatorname{si}\left(x\right)$ 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 $k=0,1,2,\dots$. …
##### 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(v) Infinite Sequences and Series
For $z$ in $\left|z-z_{0}\right|\leq\rho$ ($), the convergence is absolute and uniform. …
1.9.64 $\left|z_{m,n}-z\right|<\epsilon$
##### 6: 1.10 Functions of a Complex Variable
1.10.2 ${\mathrm{e}}^{z}=1+\frac{z}{1!}+\frac{z^{2}}{2!}+\cdots,$ $\left|z\right|<\infty$,
1.10.3 $\ln\left(1+z\right)=z-\frac{z^{2}}{2}+\frac{z^{3}}{3}-\cdots,$ $\left|z\right|<1$,
###### §1.10(ix) Infinite Products
The convergence of the product is absolute if $\prod^{\infty}_{n=1}(1+\left|a_{n}\right|)$ converges. …
1.10.26 $F(x;z)=\sum^{\infty}_{n=0}p_{n}(x)z^{n},$ $\left|z\right|.
##### 7: 1.4 Calculus of One Variable
1.4.2 $\left|f(c+\alpha)-f(c)\right|<\epsilon,$
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 $-\sum^{\infty}_{n=-\infty}\mathrm{i}(\operatorname{sign}n)F(n)r^{\left|n\right% |}{\mathrm{e}}^{\mathrm{i}n\theta};$
1.15.30 $\lim_{\epsilon\to 0+}\int^{\infty}_{-\infty}{\mathrm{e}}^{-\epsilon\left|t% \right|}f(t)\,\mathrm{d}t=L.$
1.15.35 $\int_{\left|x\right|\geq\delta}P(x,y)\,\mathrm{d}x\to 0,$ as $y\to 0$.
1.15.38 $\lim_{y\to 0+}\int^{\infty}_{-\infty}\left|h(x,y)-f(x)\right|\,\mathrm{d}x=0.$
##### 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 $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\,\mathrm{d}t$ (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 $x\to\infty$.