.新宝2世界杯结束『世界杯佣金分红55%，咨询专员：@ky975』.tbx-k2q1w9-2022年11月29日4时59分17秒

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11: 4.19 Maclaurin Series and Laurent Series

… ►

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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4.19.5 $$\sec z=1+\frac{z^2}{2}+\frac{5}{24}{z}^4+\frac{61}{720}{z}^6+\mathrm{\cdots }+\frac{{(-1)}^n{E}_{2n}}{(2n)!}{z}^{2n}+\mathrm{\cdots },$$ ,

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sec z$: secant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.69

Referenced by:

§24.1, §24.19(i), §24.2(ii)

Permalink:

http://dlmf.nist.gov/4.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

… ►

4.19.8 $$\ln \left(\cos z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{2}^{2n-1}(2^{2n}-1)B_{2n}}{n(2n)!}{z}^{2n},$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.72

Permalink:

http://dlmf.nist.gov/4.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.9 $$\ln \left(\frac{\tan z}{z}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n-1}-1)B_{2n}}{n(2n)!}{z}^{2n},$$ .

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\tan z$: tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.73

Referenced by:

§4.19, §4.33

Permalink:

http://dlmf.nist.gov/4.19.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

12: 19.5 Maclaurin and Related Expansions

… ►where ${}_2{}^{}F_1^{}$ is the Gauss hypergeometric function (§§15.1 and 15.2(i)). … ►

19.5.4_1 $$F(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.8))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

►

19.5.4_2 $$E(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},-\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.9))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

… ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►For series expansions of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when see Erdélyi et al. (1953b, §13.6(9)). …

13: 4.33 Maclaurin Series and Laurent Series

… ►

4.33.2 $$\cosh z=1+\frac{z^2}{2!}+\frac{z^4}{4!}+\mathrm{\cdots }.$$

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Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.5.63

Permalink:

http://dlmf.nist.gov/4.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.33 and Ch.4

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4.33.3 $$\tanh z=z-\frac{z^3}{3}+\frac{2}{15}{z}^5-\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ .

ⓘ

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

►For $B_{2n}$ see §24.2(i). …

14: 4.22 Infinite Products and Partial Fractions

… ►

4.22.1 $$\sin z=z\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{n^2{\pi }^2}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.89

Permalink:

http://dlmf.nist.gov/4.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.2 $$\cos z=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.90

Permalink:

http://dlmf.nist.gov/4.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

… ►

4.22.3 $$\cot z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2-n^2{\pi }^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.91

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.4 $${\csc }^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi )}^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.92

Permalink:

http://dlmf.nist.gov/4.22.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

►

4.22.5 $$\csc z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2-n^2{\pi }^2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.93

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E5

Encodings:

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See also:

Annotations for §4.22 and Ch.4

15: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.2 $$\arccos z={(2(1-z))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}(2n+1)n!}{(1-z)}^n\right),$$ $|1-z|\le 2$.

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Symbols:

$!$: factorial (as in $n!$), $\arccos z$: arccosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.4.41 (which is stated differently and the constraint on $z$ is more restrictive.)

Permalink:

http://dlmf.nist.gov/4.24.E2

Encodings:

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See also:

Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.5 $$\arctan z=\frac{z}{z^2+1}\left(1+\frac{2}{3}\frac{z^2}{1+z^2}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{1+z^2}\right)}^2+\mathrm{\cdots }\right),$$ $\mathrm{\Re }\left(z^2\right)>-\frac{1}{2}$,

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Symbols:

$\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.42 (has an error in the conditions on $z$.)

Permalink:

http://dlmf.nist.gov/4.24.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.6 $$x^2-y^2=-\frac{1}{2}.$$

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Symbols:

$x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.24.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.14 $$\mathrm{Arccos}u\pm \mathrm{Arccos}v=\mathrm{Arccos}\left(uv\mp {((1-u^2)(1-v^2))}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function

A&S Ref:

4.4.33

Permalink:

http://dlmf.nist.gov/4.24.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

… ►

4.24.16 $$\mathrm{Arcsin}u\pm \mathrm{Arccos}v=\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)=\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.35

Permalink:

http://dlmf.nist.gov/4.24.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

…

16: 30.9 Asymptotic Approximations and Expansions

… ►

§30.9(i) Prolate Spheroidal Wave Functions

►As ${\gamma }^2\to +\mathrm{\infty }$, with $q=2(n-m)+1$, … ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982). …

17: 24.20 Tables

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of ${\sum }_{k=1}^m{k}^n$, $m=1(1)100$, $n=1(1)10$; ${\sum }_{k=1}^{\mathrm{\infty }}{k}^{-n}$, ${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}{k}^{-n}$, ${\sum }_{k=0}^{\mathrm{\infty }}{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 20D; ${\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 18D. ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. In Wagstaff (2002) these results are extended to $n=60(2)152$ and $n=40(2)88$, respectively, with further complete and partial factorizations listed up to $n=300$ and $n=200$, respectively. ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

18: 34.1 Special Notation

… ► ►►

$2j_1,2j_2,2j_3,2l_1,2l_2,2l_3$	nonnegative integers.
…

… ►

►

… ►An often used alternative to the $\mathit{3}j$ symbol is the Clebsch–Gordan coefficient ►

34.1.1

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Symbols:

: $\mathit{3}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Sources:

Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))

Permalink:

http://dlmf.nist.gov/34.1.E1

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3}j$. The sign of $m_3$ was changed for clarity.

See also:

Annotations for §34.1 and Ch.34

…

19: 24.12 Zeros

… ►In the interval $0\le x\le 1$ the only zeros of $B_{2n+1}\left(x\right)$, $n=1,2,\mathrm{\dots }$, are $0,\frac{1}{2},1$, and the only zeros of $B_{2n}\left(x\right)-B_{2n}$, $n=1,2,\mathrm{\dots }$, are $0,1$. … ►When $n(\ge 2)$ is even … ►When $n$ is odd $y_1^{(n)}=\frac{1}{2}$, … ► $B_n\left(x\right)$, $n=1,2,\mathrm{\dots }$, has no multiple zeros. The only polynomial $E_n\left(x\right)$ with multiple zeros is $E_5\left(x\right)=(x-\frac{1}{2}){(x^2-x-1)}^2$.

20: 26.16 Multiset Permutations

… ►Let $S=\{1^{a_1},2^{a_2},\mathrm{\dots },n^{a_n}\}$ be the multiset that has $a_j$ copies of $j$, $1\le j\le n$. …The number of elements in ${𝔖}_S$ is the multinomial coefficient (§26.4) $\left(\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right)$. … ►The definitions of inversion number and major index can be extended to permutations of a multiset such as $351322453154\in {𝔖}_{\{1^2,2^2,3^3,4^2,5^3\}}$. Thus $inv(351322453154)=4+8+0+3+1+1+2+3+1+0+1=24$, and $maj(351322453154)=2+4+8+9+11=34.$ … ►and again with $S=\{1^{a_1},2^{a_2},\mathrm{\dots },n^{a_n}\}$ we have …