宁波驾照是国际驾照【假证加微aptao168】7Bz

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21: 4.19 Maclaurin Series and Laurent Series

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4.19.1 $$\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\mathrm{\cdots },$$

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Symbols:

$!$: factorial (as in $n!$), $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.65

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.3 $$\tan z=z+\frac{z^3}{3}+\frac{2}{15}{z}^5+\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tan z$: tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.67

Referenced by:

§4.19

Permalink:

http://dlmf.nist.gov/4.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.4 $$\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}{z}^3+\frac{31}{15120}{z}^5+\mathrm{\cdots }+\frac{{(-1)}^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

A&S Ref:

4.3.68

Referenced by:

§4.33

Permalink:

http://dlmf.nist.gov/4.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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22: 4.33 Maclaurin Series and Laurent Series

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4.33.3 $$\tanh z=z-\frac{z^3}{3}+\frac{2}{15}{z}^5-\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ .

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tanh z$: hyperbolic tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.64

Permalink:

http://dlmf.nist.gov/4.33.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.33 and Ch.4

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23: 10.76 Approximations

… ►Piessens (1984a, 1990), Piessens and Ahmed (1986), Németh (1992, Chapter 7). …

24: 13.30 Tables

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Slater (1960) tabulates $M(a,b,x)$ for $a=-1(.1)1$, $b=0.1(.1)1$, and $x=0.1(.1)10$, 7–9S; $M(a,b,1)$ for $a=-11(.2)2$ and $b=-4(.2)1$, 7D; the smallest positive $x$-zero of $M(a,b,x)$ for $a=-4(.1)-0.1$ and $b=0.1(.1)2.5$, 7D.

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25: 12.4 Power-Series Expansions

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12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

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Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.6 $$u_2(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(z+(a-\frac{3}{2})\frac{z^3}{3!}+(a-\frac{3}{2})(a-\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_2(a,z)$: solution

A&S Ref:

19.2.4 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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26: 18.13 Continued Fractions

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18.13.3 $$\frac{a_1}{x+}\frac{-\frac{1}{2}}{\frac{3}{2}x+}\frac{-\frac{2}{3}}{\frac{5}{3}x+}\frac{-\frac{3}{4}}{\frac{7}{4}x+}\mathrm{\cdots },$$

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Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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18.13.4 $$\frac{a_1}{1-x+}\frac{-\frac{1}{2}}{\frac{1}{2}(3-x)+}\frac{-\frac{2}{3}}{\frac{1}{3}(5-x)+}\frac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}\mathrm{\cdots },$$

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Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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27: 26.2 Basic Definitions

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Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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5	7	22	1002	39	31185
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7	15	24	1575	41	44583
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28: 26.6 Other Lattice Path Numbers

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Table 26.6.1: Delannoy numbers $D(m,n)$.

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$m$	$n$
	0	1	2	3	4	5	6	7	8	9	10
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1	1	3	5	7	9	11	13	15	17	19	21
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3	1	7	25	63	129	231	377	575	833	1159	1561
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Table 26.6.3: Narayana numbers $N(n,k)$.

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$n$	$k$
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7	0	1	21	105	175	105	21	1
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Table 26.6.4: Schröder numbers $r(n)$.

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$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
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$3$	$22$	$7$	$8558$	$11$	$\mathrm{52\hspace{0.33em}93446}$	$15$	$\mathrm{39376\hspace{0.33em}03038}$	$19$	$\mathrm{323\hspace{0.33em}67243\hspace{0.33em}17174}$

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29: 30.3 Eigenvalues

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30.3.10 $${\mathrm{\ell }}_6=(4m^2-1)\left(\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n-1)(2n+1){(2n+3)}^5(2n+5)(2n+7)}-\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-5)(2n-3){(2n-1)}^5(2n+1)(2n+3)}\right),$$

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Symbols:

$m$: nonnegative integer, $n\ge m$: integer degree and ${\mathrm{\ell }}_j$: coefficients

Permalink:

http://dlmf.nist.gov/30.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §30.3(iv), §30.3 and Ch.30

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$$A=\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{{(2n-5)}^2(2n-3){(2n-1)}^7(2n+1){(2n+3)}^2}-\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{{(2n-1)}^2(2n+1){(2n+3)}^7(2n+5){(2n+7)}^2},$$

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$$B=\frac{(n-m-3)(n-m-2)(n-m-1)(n-m)(n+m-3)(n+m-2)(n+m-1)(n+m)}{(2n-7){(2n-5)}^2{(2n-3)}^3{(2n-1)}^4(2n+1)}-\frac{(n-m+1)(n-m+2)(n-m+3)(n-m+4)(n+m+1)(n+m+2)(n+m+3)(n+m+4)}{(2n+1){(2n+3)}^4{(2n+5)}^3{(2n+7)}^2(2n+9)},$$

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$$C=\frac{{(n-m+1)}^2{(n-m+2)}^2{(n+m+1)}^2{(n+m+2)}^2}{{(2n+1)}^2{(2n+3)}^7{(2n+5)}^2}-\frac{{(n-m-1)}^2{(n-m)}^2{(n+m-1)}^2{(n+m)}^2}{{(2n-3)}^2{(2n-1)}^7{(2n+1)}^2},$$

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30: Publications

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Handbook of Mathematical Functions, in A Century of Excellence in Measurements Standards and Technology (D. R. Lide, ed.), CRC Press, pp. 135–139.

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A Special Functions Handbook for the Digital Age, Notices of the American Mathematical Society 58, 7 (2011), pp. 905–911.

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