哪里能买到安蒂奥克学院文凭毕业证【假证加微aptao168】8e18Buz

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21: 36.7 Zeros

… ►Inside the cusp, that is, for , the zeros form pairs lying in curved rows. … ►Just outside the cusp, that is, for $x^2>8{|y|}^3/27$, there is a single row of zeros on each side. … ►

$x_n=\pm {\left({\displaystyle \frac{8}{27}}\right)}^{1/2}{|y_n|}^{3/2}(1+{\xi }_n),$

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$y_n=-{\left({\displaystyle \frac{3\pi (8n+5)}{9+8{\xi }_n}}\right)}^{1/2},$

… ►

36.7.3 $$\frac{3\pi (8n+5)}{9+8{\xi }_n}{\xi }_n^{3/2}=\frac{27}{16}{\left(\frac{3}{2}\right)}^{1/2}\left(\ln \left(\frac{1}{{\xi }_n}\right)+3\ln \left(\frac{3}{2}\right)\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $m$: integer

Referenced by:

§36.7(ii)

Permalink:

http://dlmf.nist.gov/36.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.7(ii), §36.7 and Ch.36

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22: 5.16 Sums

… ►

5.16.1 $$\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k{\psi }^{\prime }\left(k\right)=-\frac{{\pi }^2}{8},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

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23: 13.27 Mathematical Applications

… ►Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. …

24: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

25: 34.14 Tables

… ►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. …

26: 23.20 Mathematical Applications

… ►Given $P$, calculate $2P$, $4P$, $8P$ by doubling as above. …If any of $2P$, $4P$, $8P$ is not an integer, then the point has infinite order. Otherwise observe any equalities between $P$, $2P$, $4P$, $8P$, and their negatives. The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from $2P=-P$, $4P=-P$, $4P=-2P$, $8P=P$, $8P=-P$, $8P=-2P$, or $8P=-4P$. … ►

23.20.8 $$(1-u^8)(1-v^8)={(1-uv)}^8,$$ $p=7$.

ⓘ

Symbols:

$u$ , $v$ and $p$: degree

Permalink:

http://dlmf.nist.gov/23.20.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.20(iv), §23.20 and Ch.23

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27: 27.2 Functions

… ►

Table 27.2.1: Primes.

► ►►►

$n$	$p_n$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…
8	19	61	107	163	223	271	337	397	457	521
…

►

►

Table 27.2.2: Functions related to division.

► ►►►►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
4	2	3	7	17	16	2	18	30	8	8	72	43	42	2	44
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

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28: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

… ►

$F_{\mathrm{\ell }}(\eta ,\rho )\sim {\displaystyle \frac{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{{(2\eta )}^{\mathrm{\ell }+1}}}{(2\eta \rho )}^{1/2}{I}_{2\mathrm{\ell }+1}\left({(8\eta \rho )}^{1/2}\right),$

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$G_{\mathrm{\ell }}(\eta ,\rho )\sim {\displaystyle \frac{2{(2\eta )}^{\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}}{(2\eta \rho )}^{1/2}{K}_{2\mathrm{\ell }+1}\left({(8\eta \rho )}^{1/2}\right).$

… ►

$$F_0(\eta ,\rho )\sim {\mathrm{e}}^{-\pi \eta }{(\pi \rho )}^{1/2}{I}_1\left({(8\eta \rho )}^{1/2}\right),$$

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$$G_0(\eta ,\rho )\sim 2{\mathrm{e}}^{\pi \eta }{\left(\rho /\pi \right)}^{1/2}{K}_1\left({(8\eta \rho )}^{1/2}\right),$$

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$$F_0^{\prime }(\eta ,\rho )\sim {\mathrm{e}}^{-\pi \eta }{(2\pi \eta )}^{1/2}{I}_0\left({(8\eta \rho )}^{1/2}\right),$$

…

29: 9.9 Zeros

… ►

9.9.6 $$a_k=-T\left(\frac{3}{8}\pi (4k-1)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.7 $${\mathrm{Ai}}^{\prime }\left(a_k\right)={(-1)}^{k-1}V\left(\frac{3}{8}\pi (4k-1)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.8 $$a_k^{\prime }=-U\left(\frac{3}{8}\pi (4k-3)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k^{\prime }$: $k$th zero of Airy ${\mathrm{Ai}}^{\prime }$ , $k$: nonnegative integer and $U$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.9 $$\mathrm{Ai}\left(a_k^{\prime }\right)={(-1)}^{k-1}W\left(\frac{3}{8}\pi (4k-3)\right).$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $a_k^{\prime }$: $k$th zero of Airy ${\mathrm{Ai}}^{\prime }$ , $k$: nonnegative integer and $W$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.10 $$b_k=-T\left(\frac{3}{8}\pi (4k-3)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $b_k$: $k$th zero of Airy $\mathrm{Bi}$ , $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 21))

Permalink:

http://dlmf.nist.gov/9.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

…

30: 10.61 Definitions and Basic Properties

… ►

$${\mathrm{ber}}_{\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left({\mathrm{e}}^x\cos \left(x+\frac{\pi }{8}\right)-{\mathrm{e}}^{-x}\cos \left(x-\frac{\pi }{8}\right)\right),$$

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$${\mathrm{bei}}_{\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left({\mathrm{e}}^x\sin \left(x+\frac{\pi }{8}\right)+{\mathrm{e}}^{-x}\sin \left(x-\frac{\pi }{8}\right)\right).$$

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$${\mathrm{ber}}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left({\mathrm{e}}^x\sin \left(x+\frac{\pi }{8}\right)-{\mathrm{e}}^{-x}\sin \left(x-\frac{\pi }{8}\right)\right),$$

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$${\mathrm{bei}}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left({\mathrm{e}}^x\cos \left(x+\frac{\pi }{8}\right)+{\mathrm{e}}^{-x}\cos \left(x-\frac{\pi }{8}\right)\right).$$

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10.61.11 $${\ker }_{\frac{1}{2}}\left(x\sqrt{2}\right)={\mathrm{kei}}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi }{x}}{\mathrm{e}}^{-x}\sin \left(x-\frac{\pi }{8}\right),$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function and $x$: real variable

Referenced by:

§10.61(v)

Permalink:

http://dlmf.nist.gov/10.61.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.61(v), §10.61 and Ch.10

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