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21: 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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22: 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). …

23: 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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24: 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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25: 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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26: 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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27: 10.76 Approximations

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

28: 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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29: 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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30: 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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