tangent numbers

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1: 24.15 Related Sequences of Numbers

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§24.15(ii) Tangent Numbers

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24.15.4 $$T_{2n-1}={(-1)}^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}{B}_{2n},$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$B_n$: Bernoulli numbers, $n$: integer and $T_n$: tangent numbers

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(ii), §24.15 and Ch.24

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24.15.5 $$T_{2n}=0,$$ $n=0,1,\mathrm{\dots }$.

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Symbols:

$n$: integer and $T_n$: tangent numbers

Permalink:

http://dlmf.nist.gov/24.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(ii), §24.15 and Ch.24

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Table 24.15.1: Genocchi and Tangent numbers.

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$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
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2: 24.19 Methods of Computation

… ►For example, the tangent numbers $T_n$ can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. …

3: Bibliography K

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Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 (100), pp. 663–688.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§24.15(ii), §24.19(i).

Encodings:

BibTeX

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4: 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:

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See also:

Annotations for §4.33 and Ch.4

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5: 4.19 Maclaurin Series and Laurent Series

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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:

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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

6: 21.1 Special Notation

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$g,h$	positive integers.
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$|S|$	number of elements of the set $S$.
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$a\circ b$	intersection index of $a$ and $b$, two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$.
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►Lowercase boldface letters or numbers are $g$-dimensional real or complex vectors, either row or column depending on the context. …

7: 19.11 Addition Theorems

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

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Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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8: 1.9 Calculus of a Complex Variable

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§1.9(i) Complex Numbers

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Polar Representation

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Modulus and Phase

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Powers

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Winding Number

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9: 3.5 Quadrature

… ►For the Bernoulli numbers $B_m$ see §24.2(i). … ►The $w_k$ are also known as Christoffel coefficients or Christoffel numbers and they are all positive. The remainder is given by … ►

3.5.45 $$\mathrm{erfc}\lambda =\frac{{\mathrm{e}}^{-{\lambda }^2}}{2\pi }{\int }_{-\pi }^{\pi }{\mathrm{e}}^{-{\lambda }^2{\tan }^2\left(\frac{1}{2}\theta \right)}d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\tan z$: tangent function and $\lambda$: parameter

Referenced by:

Table 3.5.20

Permalink:

http://dlmf.nist.gov/3.5.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

… ►Table 3.5.20 gives the results of applying the composite trapezoidal rule (3.5.2) with step size $h$; $n$ indicates the number of function values in the rule that are larger than ${10}^{-15}$ (we exploit the fact that the integrand is even). …

10: 19.8 Quadratic Transformations

… ►When $a_0$ and $g_0$ are positive numbers, define … ►

$${\varphi }_1=\varphi +\arctan \left(k^{\prime }\tan \varphi \right)=\arcsin \left((1+k^{\prime })\frac{\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}\right).$$

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