card证书是什么证【somewhat微aptao168】4q6DE

SearchAdvancedHelp

The terms "q6de", "aptao168" were not found.Possible alternative terms: "qcd", "caption".

(0.005 seconds)

1—10 of 519 matching pages

1: 26.18 Counting Techniques

… ►

26.18.1

ⓘ

Symbols:

$\left|A\right|$: number of elements of a finite set $A$ , $\cap$: intersection, $\setminus$: set subtraction, $\cup$: union, $j$: nonnegative integer, $n$: nonnegative integer, $A_k$: subsets and $S$: set

Permalink:

http://dlmf.nist.gov/26.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18 and Ch.26

…

2: 26.1 Special Notation

… ► ►►►

$x$	real variable.
…
$\left|A\right|$	number of elements of a finite set $A$.
…

…

3: Software Index

… ► ►►►

	Open Source													With Book						Commercial
…
4 Elementary Functions
…

… ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

…

4: 4 Elementary Functions

Chapter 4 Elementary Functions

…

5: 22.9 Cyclic Identities

… ►These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,p}^{(4)}{s}_{n,p}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }m-1$; $n\to n+1\to n+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }n-1$. … ►

22.9.17 $$d_{1,4}^{(2)}{d}_{2,4}^{(2)}{d}_{3,4}^{(2)}\pm d_{2,4}^{(2)}{d}_{3,4}^{(2)}{d}_{4,4}^{(2)}+d_{3,4}^{(2)}{d}_{4,4}^{(2)}{d}_{1,4}^{(2)}\pm d_{4,4}^{(2)}{d}_{1,4}^{(2)}{d}_{2,4}^{(2)}=k^{\prime }\left(\pm d_{1,4}^{(2)}+d_{2,4}^{(2)}\pm d_{3,4}^{(2)}+d_{4,4}^{(2)}\right),$$

ⓘ

Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

►

22.9.18 $${\left(d_{1,4}^{(2)}\right)}^2{d}_{3,4}^{(2)}\pm {\left(d_{2,4}^{(2)}\right)}^2{d}_{4,4}^{(2)}+{\left(d_{3,4}^{(2)}\right)}^2{d}_{1,4}^{(2)}\pm {\left(d_{4,4}^{(2)}\right)}^2{d}_{2,4}^{(2)}=k^{\prime }\left(d_{1,4}^{(2)}\pm d_{2,4}^{(2)}+d_{3,4}^{(2)}\pm d_{4,4}^{(2)}\right),$$

ⓘ

Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

… ►

§22.9(iv) Typical Identities of Rank 4

… ►

22.9.23 $$s_{1,3}^{(4)}{d}_{1,3}^{(4)}{c}_{2,3}^{(4)}{c}_{3,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}{c}_{3,3}^{(4)}{c}_{1,3}^{(4)}+s_{3,3}^{(4)}{d}_{3,3}^{(4)}{c}_{1,3}^{(4)}{c}_{2,3}^{(4)}=\frac{{\kappa }^2}{1-{\kappa }^2}\left(s_{1,3}^{(4)}{d}_{1,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}\right).$$

ⓘ

Symbols:

$s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

…

6: 6.3 Graphics

… ►

►

►

Figure 6.3.3: $|E_1\left(x+\mathrm{i}y\right)|$, $-4\le x\le 4$, $-4\le y\le 4$. … Magnify 3D Help

7: 9.4 Maclaurin Series

… ►

9.4.1 $$\mathrm{Ai}\left(z\right)=\mathrm{Ai}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Ai}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Source:

Olver (1997b, p. 54)

A&S Ref:

10.4.2 (with 10.4.4 and 10.4.5)

Permalink:

http://dlmf.nist.gov/9.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.2 $${\mathrm{Ai}}^{\prime }\left(z\right)={\mathrm{Ai}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Ai}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.3 $$\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Bi}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

A&S Ref:

10.4.3 (with 10.4.4 and 10.4.5)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.4 $${\mathrm{Bi}}^{\prime }\left(z\right)={\mathrm{Bi}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Bi}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right).$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

8: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

►For properties of Struve functions of argument $x{\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$ see McLachlan and Meyers (1936).

9: 22.10 Maclaurin Series

… ►

22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.2 $$\mathrm{cn}(z,k)=1-\frac{z^2}{2!}+\left(1+4k^2\right)\frac{z^4}{4!}-\left(1+44k^2+16k^4\right)\frac{z^6}{6!}+O\left(z^8\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.2

Permalink:

http://dlmf.nist.gov/22.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.3 $$\mathrm{dn}(z,k)=1-k^2\frac{z^2}{2!}+k^2\left(4+k^2\right)\frac{z^4}{4!}-k^2\left(16+44k^2+k^4\right)\frac{z^6}{6!}+O\left(z^8\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.3

Permalink:

http://dlmf.nist.gov/22.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

… ►

22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

►

22.10.5 $$\mathrm{cn}(z,k)=\cos z+\frac{k^2}{4}(z-\sin z\cos z)\sin z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.2

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

…

10: 4.43 Cubic Equations

… ►

$A={\left(-\frac{4}{3}p\right)}^{1/2},$

►

$B={\left(\frac{4}{3}p\right)}^{1/2}.$

… ►

$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

►

$A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=-4q/A^3$, when , , and $4p^3+27q^2>0$.

►

$B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, when $p>0$.

…