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31—40 of 165 matching pages

31: 10.76 Approximations

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

32: 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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33: 18.41 Tables

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34: 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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4	1	9	41	129	321	681	1289	2241	3649	5641	8361
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9	1	19	181	1159	5641	22363	75517	2 24143	5 98417	14 62563	33 17445
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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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$1$	$2$	$5$	$394$	$9$	$\mathrm{2\hspace{0.33em}06098}$	$13$	$\mathrm{1420\hspace{0.33em}78746}$	$17$	$\mathrm{11\hspace{0.33em}18180\hspace{0.33em}26018}$
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35: 28.16 Asymptotic Expansions for Large $q$

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28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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36: 28.15 Expansions for Small $q$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

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37: About the Project

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Figure 1: The Editors and 9 of the 10 Associate Editors of the DLMF Project (photo taken at 3rd Editors Meeting, April, 2001). …

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38: Staff

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Frank W. J. Olver, University of Maryland and NIST, Chaps. 1, 2, 4, 9, 10

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Diego Dominici, State University of New York at New Paltz, for Chaps. 9, 10 (deceased)

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39: 7.9 Continued Fractions

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7.9.2 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{2z}{2z^2+1-}\frac{1\cdot 2}{2z^2+5-}\frac{3\cdot 4}{2z^2+9-}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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40: 36.7 Zeros

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

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

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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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$\Delta z={\displaystyle \frac{9\pi }{2z_n^2}},$

… ►The rings are almost circular (radii close to $(\Delta x)/9$ and varying by less than 1%), and almost flat (deviating from the planes $z_n$ by at most $(\Delta z)/36$). …