# minimum

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##### 1: 3.1 Arithmetics and Error Measures
Let $E_{{\rm min}}\leq E\leq E_{{\rm max}}$ with $E_{{\rm min}}<0$ and $E_{{\rm max}}>0$. …The integers $p$, $E_{{\rm min}}$, and $E_{{\rm max}}$ are characteristics of the machine. …Underflow (overflow) after computing $x\neq 0$ occurs when $|x|$ is smaller (larger) than $N_{{\rm min}}$ ($N_{{\rm max}}$). … In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ($N=32$, $p=24$, $E_{{\rm min}}=-126$, $E_{{\rm max}}=127$), binary64 (previously double precision) ($N=64$, $p=53$, $E_{{\rm min}}=-1022$, $E_{{\rm max}}=1023$) and binary128 (previously quad precision) ($N=128$, $p=113$, $E_{{\rm min}}=-16382$, $E_{{\rm max}}=16383$) are as in Figure 3.1.1. … $N_{{\rm min}}\leq x\leq N_{{\rm max}}$, and …
##### 2: 15.14 Integrals
15.14.1 $\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\,\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},$ $\min(\Re a,\Re b)>\Re s>0$.
##### 3: 18.37 Classical OP’s in Two or More Variables
18.37.1 $R^{(\alpha)}_{m,n}\left(r{\mathrm{e}}^{\mathrm{i}\theta}\right)={\mathrm{e}}^{% \mathrm{i}(m-n)\theta}r^{|m-n|}\frac{P^{(\alpha,|m-n|)}_{\min(m,n)}\left(2r^{2% }-1\right)}{P^{(\alpha,|m-n|)}_{\min(m,n)}\left(1\right)},$ $r\geq 0$, $\theta\in\mathbb{R}$, $\alpha>-1$.
18.37.3 $R^{(\alpha)}_{m,n}\left(z\right)=\sum_{j=0}^{\min(m,n)}c_{j}z^{m-j}{\overline{% z}}^{n-j},$
18.37.4 $\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)(x-iy)% ^{m-j}(x+iy)^{n-j}\*(1-x^{2}-y^{2})^{\alpha}\,\mathrm{d}x\,\mathrm{d}y=0,$ $j=1,2,\dots,\min(m,n)$;
18.37.6 $R^{(\alpha)}_{m,n}\left(z\right)=\sum_{j=0}^{\min(m,n)}\frac{(-1)^{j}{\left(% \alpha+1\right)_{m+n-j}}{\left(-m\right)_{j}}{\left(-n\right)_{j}}}{{\left(% \alpha+1\right)_{m}}{\left(\alpha+1\right)_{n}}j!}\*z^{m-j}\*{\overline{z}}^{n% -j}.$
##### 4: 1.4 Calculus of One Variable
###### Maxima and Minima
A necessary condition that a differentiable function $f(x)$ has a local maximum (minimum) at $x=c$, that is, $f(x)\leq f(c)$, ($f(x)\geq f(c)$) in a neighborhood $c-\delta\leq x\leq c+\delta$ ($\delta>0$) of $c$, is $f^{\prime}(c)=0$. …
###### §1.4(vii) Maxima and Minima
If $f(x)$ is twice-differentiable, and if also $f^{\prime}(x_{0})=0$ and $f^{\prime\prime}(x_{0})<0$ ($>0$), then $x=x_{0}$ is a local maximum (minimum) (§1.4(iii)) of $f(x)$. The overall maximum (minimum) of $f(x)$ on $[a,b]$ will either be at a local maximum (minimum) or at one of the end points $a$ or $b$. …
##### 5: 5.14 Multidimensional Integrals
provided that $\Re a$, $\Re b>0$, $\Re c>-\min(1/n,\Re a/(n-1),\Re b/(n-1))$. … when $\Re a>0$, $\Re c>-\min(1/n,\Re a/(n-1))$. …
##### 6: 20.6 Power Series
where $z_{m,n}$ is given by (20.2.5) and the minimum is for $m,n\in\mathbb{Z}$, except $m=n=0$. …
##### 7: 19.27 Asymptotic Approximations and Expansions
19.27.13 $R_{J}\left(x,y,z,p\right)=\frac{3}{2\sqrt{z}p}\left(\ln\left(\frac{8z}{a+g}% \right)-2R_{C}\left(1,\frac{p}{z}\right)+O\left(\left(\frac{a}{z}+\frac{a}{p}% \right)\ln\frac{p}{a}\right)\right),$ $\max(x,y)/\min(z,p)\to 0$.
19.27.14 $R_{J}\left(x,y,z,p\right)=\frac{3}{\sqrt{yz}}R_{C}\left(x,p\right)-\frac{6}{yz% }R_{G}\left(0,y,z\right)+O\left(\frac{\sqrt{x+2p}}{yz}\right),$ $\max(x,p)/\min(y,z)\to 0$.
19.27.15 $R_{J}\left(x,y,z,p\right)=R_{J}\left(0,y,z,p\right)-\frac{3\sqrt{x}}{hp}\left(% 1+O\left(\left(\frac{b}{h}+\frac{h}{p}\right)\sqrt{\frac{x}{h}}\right)\right),$ $x/\min(y,z,p)\to 0$.
##### 8: 19.9 Inequalities
19.9.7 $(1-\tfrac{1}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(E\left(k\right)-{k^{\prime}}^% {2}K\left(k\right))<\min((k^{\prime})^{-1/4},4/\pi),$
19.9.11 $\phi\leq F\left(\phi,k\right)\leq\min(\phi/\Delta,{\operatorname{gd}^{-1}}% \left(\phi\right)),$
19.9.13 $\Pi\left(\phi,\alpha^{2},0\right)\leq\Pi\left(\phi,\alpha^{2},k\right)\leq\min% (\Pi\left(\phi,\alpha^{2},0\right)/\Delta,\Pi\left(\phi,\alpha^{2},1\right)).$
##### 9: 14.21 Definitions and Basic Properties
The generating function expansions (14.7.19) (with $\mathsf{P}$ replaced by $P$) and (14.7.22) apply when $|h|<\min\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$; (14.7.21) (with $\mathsf{P}$ replaced by $P$) applies when $|h|>\max\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$.
##### 10: 19.34 Mutual Inductance of Coaxial Circles
is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. …