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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: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

… ►

37.6.3

ⓘ

Defines:

$S_{m,n}(z,\overline{z})$: two-variable complex circular Hermite polynomial

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $|z|$: absolute value (modulus) for complex numbers, $\overline{z}$: complex conjugate, $!$: factorial (as in $n!$), $n$: nonnegative integer, $z$: complex number and $m$: nonnegative integer

Sources:

Itô (1952, §12); Intissar and Intissar (2006, Proposition 2.1(i)($\beta$))

Referenced by:

§37.12(iv), §37.6(ii), §37.6(iii)

Permalink:

http://dlmf.nist.gov/37.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(i), §37.6 and Ch.37

… ►

37.6.5 $$S_{m,n}(z,\overline{z})=\sum _{j=0}^{\min \{m,n\}}{(-1)}^{j}j!\left(\genfrac{}{}{0.0pt}{}{m}{j}\right)\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)z^{m-j}{\overline{z}}^{n-j}=z^m{\overline{z}}^n{}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-m,-n}{-};-{|z|}^{-2}).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $|z|$: absolute value (modulus) for complex numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $S_{m,n}(z,\overline{z})$: two-variable complex circular Hermite polynomial, $\overline{z}$: complex conjugate, $!$: factorial (as in $n!$), $n$: nonnegative integer, $z$: complex number, $j$: nonnegative integer and $m$: nonnegative integer

Proved:

Intissar and Intissar (2006, Proof of Proposition 2.1); For the first equality in (37.6.5).(proved)

Referenced by:

(37.6.5), (37.6.7)

Permalink:

http://dlmf.nist.gov/37.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(i), §37.6(i), §37.6 and Ch.37

… ►

37.6.9 $$S_{m,n}(z,\overline{z})={(-1)}^{m+n}{\mathrm{e}}^{{|z|}^2}{D}_{\overline{z}}^m{D}_z^n{\mathrm{e}}^{-{|z|}^2}.$$

ⓘ

Symbols:

$|z|$: absolute value (modulus) for complex numbers, $S_{m,n}(z,\overline{z})$: two-variable complex circular Hermite polynomial, $\overline{z}$: complex conjugate, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer, $z$: complex number and $m$: nonnegative integer

Proved:

Intissar and Intissar (2006, Proposition 2.1)(proved)

Referenced by:

(37.6.10), (37.6.8)

Permalink:

http://dlmf.nist.gov/37.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(i), §37.6(i), §37.6 and Ch.37

…

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 $k=0,1,2,\mathrm{\dots }$. … ►In (6.13.2), the remainder after $n$th terms does not exceed the $(n+1)$th term in absolute value and has the same sign. …

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

ⓘ

Defines:

${ϵ}_a$: absolute error (locally)

A&S Ref:

3.5.1 (in absolute value)

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

…

5: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

… ►

37.4.11

ⓘ

Defines:

$R_{m,n}^{\alpha }(z,\overline{z})$: two-variable complex disk polynomial

Symbols:

$R_n^{(\alpha ,\beta )}\left(x\right)$: standardized Jacobi polynomial, $|z|$: absolute value (modulus) for complex numbers, $\overline{z}$: complex conjugate, $n$: nonnegative integer, $z$: complex number, $m$: nonnegative integer, $P_{k,n}$: two-variable ridge polynomial on $𝔻$ and $\alpha$: real parameter

Source:

Dunkl and Xu (2014, (2.6.15), (2.6.16)); There the notation is $P_{m,n}^{\alpha }(z,\overline{z})$.

Referenced by:

§37.11(iv), §37.12(iv), §37.15(iii), (37.4.30), §37.4(iv), §37.4(iv), §37.4(vi)

Permalink:

http://dlmf.nist.gov/37.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §37.4(iv), §37.4(iv), §37.4 and Ch.37

… ►

37.4.17 $$\left|R_{m,n}^{\alpha }(z,\overline{z})\right|\le 1,$$ $|z|\le 1$, $\alpha \ge 0$.

ⓘ

Symbols:

$|z|$: absolute value (modulus) for complex numbers, $R_{m,n}^{\alpha }(z,\overline{z})$: two-variable complex disk polynomial, $\overline{z}$: complex conjugate, $n$: nonnegative integer, $z$: complex number, $m$: nonnegative integer and $\alpha$: real parameter

Proved:

Dunkl and Xu (2014, Proposition 2.6.7(i))(proved)

Permalink:

http://dlmf.nist.gov/37.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §37.4(iv), §37.4(iv), §37.4 and Ch.37

… ►

$$\left(\left(1-{|z|}^2\right)D_z-\alpha \overline{z}\right)R_{m,n}^{\alpha }(z,\overline{z})=-\alpha R_{m,n+1}^{\alpha -1}(z,\overline{z}),$$

►

$$\left(\left(1-{|z|}^2\right)D_{\overline{z}}-\alpha z\right)R_{m,n}^{\alpha }(z,\overline{z})=-\alpha R_{m+1,n}^{\alpha -1}(z,\overline{z}).$$

… ►

37.4.21 $$R_{m,n}^{\alpha }(z,\overline{z})=\frac{{(-1)}^{m+n}}{{\left(\alpha +1\right)}_{m+n}}{\left(1-{|z|}^2\right)}^{-\alpha }{D}_{\overline{z}}^m{D}_z^n{\left(1-{|z|}^2\right)}^{\alpha +m+n}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $|z|$: absolute value (modulus) for complex numbers, $R_{m,n}^{\alpha }(z,\overline{z})$: two-variable complex disk polynomial, $\overline{z}$: complex conjugate, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer, $z$: complex number, $m$: nonnegative integer and $\alpha$: real parameter

Proved:

Wünsche (2005, (2.6))(proved)

Referenced by:

(37.4.20)

Permalink:

http://dlmf.nist.gov/37.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §37.4(iv), §37.4(iv), §37.4 and Ch.37

…

6: 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. …

7: 1.9 Calculus of a Complex Variable

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

… ►

§1.9(v) Infinite Sequences and Series

… ►

1.9.47

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

Encodings:

pMML, png, TeX

See also:

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

…

8: 1.10 Functions of a Complex Variable

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

Encodings:

pMML, png, TeX

See also:

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

Encodings:

pMML, png, TeX

See also:

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

…

9: 37.16 Orthogonal Polynomials on the Hyperoctant

… ►

37.16.7 $${𝐏}_z^{𝜶}(𝐱,𝐲)=\sum _{n=0}^{\mathrm{\infty }}{𝐑}_n^{𝜶}(𝐱,𝐲)z^n={(1-z)}^{-1}\exp \left(\frac{z(|𝐱|+|𝐲|)}{z-1}\right)\prod _{\mathrm{\ell }=1}^d\frac{\mathrm{\Gamma }\left({\alpha }_{\mathrm{\ell }}+1\right)}{{\left(x_{\mathrm{\ell }}{y}_{\mathrm{\ell }}z\right)}^{\frac{1}{2}{\alpha }_{\mathrm{\ell }}}}{I}_{{\alpha }_{\mathrm{\ell }}}\left(\frac{2\sqrt{x_{\mathrm{\ell }}{y}_{\mathrm{\ell }}z}}{1-z}\right),$$ , $𝐱,𝐲\in {ℝ}_{+}^d$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $|z|$: absolute value (modulus) for complex numbers, $\in$: element of, $\exp z$: exponential function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $ℝ$: real line, $n$: nonnegative integer, $z$: complex number, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ , ${𝐏}_z$: Poisson kernel on ${𝒱}_n^d$ , ${ℝ}_{+}^d$: hyperoctant, $x$: real variable and $y$: real variable

Proved:

Dunkl and Xu (2014, Proposition 8.4.8); There set $V_{\kappa }=\mathrm{id}$.(proved)

Permalink:

http://dlmf.nist.gov/37.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16(ii), §37.16 and Ch.37

…

10: 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

Encodings:

pMML, png, TeX

See also:

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. …