# tangent numbers

(0.001 seconds)

## 1—10 of 24 matching pages

##### 1: 24.15 Related Sequences of Numbers
###### §24.15(ii) TangentNumbers
24.15.4 $T_{2n-1}=(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}B_{2n},$ $n=1,2,\dots$,
24.15.5 $T_{2n}=0,$ $n=0,1,\dots$.
##### 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
• D. E. Knuth and T. J. Buckholtz (1967) Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 (100), pp. 663–688.
• ##### 4: 4.33 Maclaurin Series and Laurent Series
4.33.3 $\tanh z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}z^{7}+\cdots+\frac{2% ^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$.
##### 5: 4.19 Maclaurin Series and Laurent Series
4.19.3 $\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$,
4.19.9 $\ln\left(\frac{\tan z}{z}\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{% 2n-1}-1)B_{2n}}{n(2n)!}z^{2n},$ $|z|<\frac{1}{2}\pi$.
##### 6: 21.1 Special Notation
 $g,h$ positive integers. … number of elements of the set $S$. … 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 $-1$. …
Lowercase boldface letters or numbers are $g$-dimensional real or complex vectors, either row or column depending on the context. …
##### 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 $\operatorname{erfc}\lambda=\frac{e^{-\lambda^{2}}}{2\pi}\int_{-\pi}^{\pi}e^{-% \lambda^{2}{\tan}^{2}\left(\frac{1}{2}\theta\right)}\mathrm{d}\theta.$
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 …
$\phi_{1}=\phi+\operatorname{arctan}\left(k^{\prime}\tan\phi\right)=% \operatorname{arcsin}\left((1+k^{\prime})\frac{\sin\phi\cos\phi}{\sqrt{1-k^{2}% {\sin}^{2}\phi}}\right).$