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1: 3.1 Arithmetics and Error Measures

… ►Let $E_{\min }\le E\le E_{\max }$ with and $E_{\max }>0$. …The integers $p$, $E_{\min }$, and $E_{\max }$ are characteristics of the machine. …Underflow (overflow) after computing $x\ne 0$ occurs when $|x|$ is smaller (larger) than $N_{\min }$ ($N_{\max }$). … ►In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ($N=32$, $p=24$, $E_{\min }=-126$, $E_{\max }=127$), binary64 (previously double precision) ($N=64$, $p=53$, $E_{\min }=-1022$, $E_{\max }=1023$) and binary128 (previously quad precision) ($N=128$, $p=113$, $E_{\min }=-16382$, $E_{\max }=16383$) are as in Figure 3.1.1. … ► $N_{\min }\le x\le N_{\max }$, and …

2: 15.14 Integrals

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15.14.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}𝐅(\genfrac{}{}{0pt}{}{a,b}{c};-x)dx=\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(a-s\right)\mathrm{\Gamma }\left(b-s\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-s\right)},$$ $\min (\mathrm{\Re }a,\mathrm{\Re }b)>\mathrm{\Re }s>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $s$: nonnegative integer, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.14

Permalink:

http://dlmf.nist.gov/15.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.14 and Ch.15

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3: 18.37 Classical OP’s in Two or More Variables

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18.37.1 $$R_{m,n}^{(\alpha )}\left(r{\mathrm{e}}^{\mathrm{i}\theta }\right)={\mathrm{e}}^{\mathrm{i}(m-n)\theta }{r}^{|m-n|}\frac{P_{\min (m,n)}^{(\alpha ,|m-n|)}\left(2r^2-1\right)}{P_{\min (m,n)}^{(\alpha ,|m-n|)}\left(1\right)},$$ $r\ge 0$, $\theta \in ℝ$, $\alpha >-1$.

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Defines:

$R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $ℝ$: real line, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.37(ii)

Permalink:

http://dlmf.nist.gov/18.37.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.3 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}{c}_j{z}^{m-j}{\overline{z}}^{n-j},$$

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Symbols:

$\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.4 $j=1,2,\mathrm{\dots },\min (m,n)$;

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Symbols:

$dx$: differential of $x$ , $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $\mathrm{i}$: imaginary unit, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.6 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}\frac{{(-1)}^j{\left(\alpha +1\right)}_{m+n-j}{\left(-m\right)}_j{\left(-n\right)}_j}{{\left(\alpha +1\right)}_m{\left(\alpha +1\right)}_{n}j!}{z}^{m-j}{\overline{z}}^{n-j}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $!$: factorial (as in $n!$), $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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4: 1.4 Calculus of One Variable

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Maxima and Minima

►A necessary condition that a differentiable function $f(x)$ has a local maximum (minimum) at $x=c$, that is, $f(x)\le f(c)$, ($f(x)\ge f(c)$) in a neighborhood $c-\delta \le x\le c+\delta$ ($\delta >0$) 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 ($>0$), 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 $\mathrm{\Re }a$, $\mathrm{\Re }b>0$, $\mathrm{\Re }c>-\min (1/n,\mathrm{\Re }a/(n-1),\mathrm{\Re }b/(n-1))$. … ►when $\mathrm{\Re }a>0$, $\mathrm{\Re }c>-\min (1/n,\mathrm{\Re }a/(n-1))$. …

6: 20.6 Power Series

… ►

20.6.1

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer and $z$: complex

Permalink:

http://dlmf.nist.gov/20.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

►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

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19.27.13 $$R_J(x,y,z,p)=\frac{3}{2\sqrt{z}p}\left(\ln \left(\frac{8z}{a+g}\right)-2R_C(1,\frac{p}{z})+O\left(\left(\frac{a}{z}+\frac{a}{p}\right)\ln \frac{p}{a}\right)\right),$$ $\max (x,y)/\min (z,p)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.12, §19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

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19.27.14 $$R_J(x,y,z,p)=\frac{3}{\sqrt{yz}}{R}_C(x,p)-\frac{6}{yz}{R}_G(0,y,z)+O\left(\frac{\sqrt{x+2p}}{yz}\right),$$ $\max (x,p)/\min (y,z)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

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19.27.15 $$R_J(x,y,z,p)=R_J(0,y,z,p)-\frac{3\sqrt{x}}{hp}\left(1+O\left(\left(\frac{b}{h}+\frac{h}{p}\right)\sqrt{\frac{x}{h}}\right)\right),$$ $x/\min (y,z,p)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $b$ and $h$

Permalink:

http://dlmf.nist.gov/19.27.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

…

8: 19.9 Inequalities

… ►

19.9.7

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

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19.9.11 $$\varphi \le F(\varphi ,k)\le \min (\varphi /\mathrm{\Delta },{\mathrm{gd}}^{-1}\left(\varphi \right)),$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.13 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,0)\le \mathrm{\Pi }(\varphi ,{\alpha }^2,k)\le \min (\mathrm{\Pi }(\varphi ,{\alpha }^2,0)/\mathrm{\Delta },\mathrm{\Pi }(\varphi ,{\alpha }^2,1)).$$

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Symbols:

$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $\mathrm{\Delta }$

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

…

9: 14.21 Definitions and Basic Properties

… ►The generating function expansions (14.7.19) (with $𝖯$ replaced by $P$) and (14.7.22) apply when ; (14.7.21) (with $𝖯$ 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. …