宁波驾照是国际驾照【假证加微aptao168】7Bz

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11: Bibliography Q

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On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).

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Notes:

In Chinese

Referenced by:

§19.9(i).

Encodings:

BibTeX

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:

ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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12: Nico M. Temme

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7 Error Functions, Dawson’s and Fresnel Integrals

… ►In November 2015, Temme was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 3, 6, 7, and 12.

13: 26.9 Integer Partitions: Restricted Number and Part Size

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Table 26.9.1: Partitions $p_k\left(n\right)$.

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$n$	$k$
	0	1	2	3	4	5	6	7	8	9	10
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5	0	1	3	5	6	7	7	7	7	7	7
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
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Figure 26.9.1: Ferrers graph of the partition $7+4+3+3+2+1$.

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14: 32.12 Asymptotic Approximations for Complex Variables

… ►See Boutroux (1913), Novokshënov (1990), Kapaev (1991), Joshi and Kruskal (1992), Kitaev (1994), Its and Kapaev (2003), and Fokas et al. (2006, Chapter 7). …

15: 23.23 Tables

… ►2 in Abramowitz and Stegun (1964) gives values of $\mathrm{\wp }\left(z\right)$, ${\mathrm{\wp }}^{\prime }\left(z\right)$, and $\zeta \left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that ${\omega }_1=1$ and ${\omega }_3=\mathrm{i}a$ (rectangular case), or ${\omega }_1=1$ and ${\omega }_3=\frac{1}{2}+\mathrm{i}a$ (rhombic case), for $a$ = 1. …

16: 24.2 Definitions and Generating Functions

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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$14$	$\frac{7}{6}$	$-\mathrm{1993\hspace{0.33em}60981}$
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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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$14$	$7$	$6$	$\mathrm{1.16666\hspace{0.33em}6667}$
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Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
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$7$	$0$	$\frac{1}{6}$	$0$	$-\frac{7}{6}$	$0$	$\frac{7}{2}$	$-\frac{7}{2}$	$1$
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Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

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	$k$
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$7$	$\frac{17}{8}$	$0$	$-\frac{21}{2}$	$0$	$\frac{35}{4}$	$0$	$-\frac{7}{2}$	$1$
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17: 3.4 Differentiation

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$$B_{-3}^7=-\frac{1}{5040}(48-56t-168t^2+140t^3+35t^4-42t^5+7t^6),$$

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$$B_{-2}^7=\frac{1}{720}(72-108t-213t^2+240t^3-10t^4-36t^5+7t^6),$$

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$$B_{-1}^7=-\frac{1}{240}(144-360t-48t^2+260t^3-45t^4-30t^5+7t^6),$$

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$$B_0^7=-\frac{1}{144}(36+392t-147t^2-224t^3+70t^4+24t^5-7t^6),$$

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$$B_1^7=\frac{1}{144}(144+216t-264t^2-156t^3+85t^4+18t^5-7t^6),$$

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18: 10.73 Physical Applications

… ►See Jackson (1999, Chapter 3, §§3.7, 3.8, 3.11, 3.13), Lamb (1932, Chapter V, §§100–102; Chapter VIII, §§186, 191–193; Chapter X, §§303, 304), Happel and Brenner (1973, Chapter 3, §3.3; Chapter 7, §7.3), Korenev (2002, Chapter 4, §43), and Gray et al. (1922, Chapter XI). … ►See Jackson (1999, Chapter 9, §9.6), Jones (1986, Chapters 7, 8), and Lord Rayleigh (1945, Vol. I, Chapter IX, §§200–211, 218, 219, 221a; Vol. II, Chapter XIII, §272a; Chapter XV, §302; Chapter XVIII; Chapter XIX, §350; Chapter XX, §357; Chapter XXI, §369). …See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►See Smith (1997, Chapter 3, §3.7; Chapter 6, §6.4), Beckmann and Spizzichino (1963, Chapter 4, §§4.2, 4.3; Chapter 5, §§5.2, 5.3; Chapter 6, §6.1; Chapter 7, §7.1.), Kerker (1969, Chapter 5, §5.6.4; Chapter 7, §7.5.6), and Bayvel and Jones (1981, Chapter 1, §§1.6.5, 1.6.6). … ►See Messiah (1961, Chapter IX, §§7–10). …

19: 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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$$=1+\frac{z}{1-}\frac{z}{2+}\frac{z}{3-}\frac{z}{2+}\frac{z}{5-}\frac{z}{2+}\frac{z}{7-}\mathrm{\cdots }$$

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20: 7.17 Inverse Error Functions

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7.17.2 $$\mathrm{inverf}x=t+\frac{1}{3}{t}^3+\frac{7}{30}{t}^5+\frac{127}{630}{t}^7+\mathrm{\cdots }=\sum _{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1},$$ ,

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Symbols:

$\mathrm{inverf}x$: inverse error function and $x$: real variable

Referenced by:

Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The equality “$={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1}$” was added to this equation.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

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7.17.3 $$\mathrm{inverfc}x\sim u^{-1/2}+a_2{u}^{3/2}+a_3{u}^{5/2}+a_4{u}^{7/2}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{inverfc}x$: inverse complementary error function, $x$: real variable, $a_{\mathrm{i}}$: coefficients and $u$: expansion variable

Referenced by:

§7.17(iii), §7.17(iii)

Permalink:

http://dlmf.nist.gov/7.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.17(iii), §7.17 and Ch.7

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