阿斯顿大学理工科文凭证书【仿证 微kaa77788】】9dm

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21: 14.33 Tables

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Abramowitz and Stegun (1964, Chapter 8) tabulates ${𝖯}_n\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 5–8D; ${𝖯}_n^{\prime }\left(x\right)$ for $n=1(1)4,9,10$, $x=0(.01)1$, 5–7D; ${𝖰}_n\left(x\right)$ and ${𝖰}_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 6–8D; $P_n\left(x\right)$ and $P_n^{\prime }\left(x\right)$ for $n=0(1)5,9,10$, $x=1(.2)10$, 6S; $Q_n\left(x\right)$ and $Q_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=1(.2)10$, 6S. (Here primes denote derivatives with respect to $x$.)

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22: 26.3 Lattice Paths: Binomial Coefficients

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Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

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$m$	$n$
	0	1	2	3	4	5	6	7	8	9	10
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9	1	9	36	84	126	126	84	36	9	1
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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

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$m$	$n$
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1	1	2	3	4	5	6	7	8	9
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8	1	9	45	165	495	1287	3003	6435	12870

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23: 4.39 Continued Fractions

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4.39.2 $$\frac{\mathrm{arcsinh}z}{\sqrt{1+z^2}}=\frac{z}{1+}\frac{1\cdot 2z^2}{3+}\frac{1\cdot 2z^2}{5+}\frac{3\cdot 4z^2}{7+}\frac{3\cdot 4z^2}{9+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.36

Permalink:

http://dlmf.nist.gov/4.39.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.39 and Ch.4

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4.39.3 $$\mathrm{arctanh}z=\frac{z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\mathrm{\cdots },$$

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Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.35

Permalink:

http://dlmf.nist.gov/4.39.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.39 and Ch.4

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24: 4.9 Continued Fractions

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4.9.1 $$\ln \left(1+z\right)=\frac{z}{1+}\frac{z}{2+}\frac{z}{3+}\frac{4z}{4+}\frac{4z}{5+}\frac{9z}{6+}\frac{9z}{7+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.39

Permalink:

http://dlmf.nist.gov/4.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

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4.9.2 $$\ln \left(\frac{1+z}{1-z}\right)=\frac{2z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\frac{16z^2}{9-}\mathrm{\cdots },$$

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Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.40

Permalink:

http://dlmf.nist.gov/4.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

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25: Diego Dominici

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9 Airy and Related Functions

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26: Mourad E. H. Ismail

… ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …

27: 28.6 Expansions for Small $q$

… ►Leading terms of the of the power series for $m=7,8,9,\mathrm{\dots }$ are: ►

28.6.14 $$\begin{array}{c}{a}_m\left(q\right)\\ b_m\left(q\right)\end{array}\}=m^2+\frac{1}{2(m^2-1)}{q}^2+\frac{5m^2+7}{32{(m^2-1)}^3(m^2-4)}{q}^4+\frac{9m^4+58m^2+29}{64{(m^2-1)}^5(m^2-4)(m^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $m$: integer and $q=h^2$: parameter

A&S Ref:

20.2.26

Referenced by:

§28.6(i), §28.6(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

… ►Numerical values of the radii of convergence ${\rho }_n^{(j)}$ of the power series (28.6.1)–(28.6.14) for $n=0,1,\mathrm{\dots },9$ are given in Table 28.6.1. … ►

28.6.22 $${\mathrm{ce}}_1(z,q)=\cos z-\frac{1}{8}q\cos 3z+\frac{1}{128}{q}^2\left(\frac{2}{3}\cos 5z-2\cos 3z-\cos z\right)-\frac{1}{1024}{q}^3\left(\frac{1}{9}\cos 7z-\frac{8}{9}\cos 5z-\frac{1}{3}\cos 3z+2\cos z\right)+\mathrm{\cdots },$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $q=h^2$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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28.6.23 $${\mathrm{se}}_1(z,q)=\sin z-\frac{1}{8}q\sin 3z+\frac{1}{128}{q}^2\left(\frac{2}{3}\sin 5z+2\sin 3z-\sin z\right)-\frac{1}{1024}{q}^3\left(\frac{1}{9}\sin 7z+\frac{8}{9}\sin 5z-\frac{1}{3}\sin 3z-2\sin z\right)+\mathrm{\cdots },$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $q=h^2$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.6.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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28: 10.76 Approximations

… ►Luke (1975, Table 9.10), Németh (1992, Chapter 9).

29: 15.3 Graphics

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Figure 15.3.1: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x),-100\le x\le 1$. Magnify

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Figure 15.3.5: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x+\mathrm{i}y),0\le x\le 2,-0.5\le y\le 0.5$. … Magnify 3D Help

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30: 18.41 Tables

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