Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

► ►►

	$\sinh\theta=a$	$\cosh\theta=a$	$\tanh\theta=a$	$\operatorname{csch}\theta=a$	$\operatorname{sech}\theta=a$	$\coth\theta=a$
…

►

25: 6.6 Power Series

… ►The series in this section converge for all finite values of

x

and

|z|

26: 9.17 Methods of Computation

… ►Although the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of

z

, they are cumbersome to use when

|z|

is large owing to slowness of convergence and cancellation. …

27: 11.13 Methods of Computation

… ►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of

z

, they are cumbersome to use when

|z|

is large owing to slowness of convergence and cancellation. …

28: 20.14 Methods of Computation

… ►In theory, starting from any value of

\tau

, a finite number of applications of the transformations

\tau\to\tau+1

and

\tau\to-1/\tau

will result in a value of

\tau

with

\Im\tau\geq\sqrt{3}/2

; see §23.18. …

29: 26.2 Basic Definitions

… ►Given a finite set

S

with permutation

\sigma

, a cycle is an ordered equivalence class of elements of

S

where

j

is equivalent to

k

if there exists an

\ell=\ell(j,k)

such that

j=\sigma^{\ell}(k)

, where

\sigma^{1}=\sigma

and

\sigma^{\ell}

is the composition of

\sigma

with

\sigma^{\ell-1}

. …

30: 26.18 Counting Techniques

… ►

26.18.1

\left|S\setminus(A_{1}\cup A_{2}\cup\cdots\cup A_{n})\right|=\left|S\right|+% \sum_{t=1}^{n}(-1)^{t}\sum_{1\leq j_{1}<j_{2}<\cdots<j_{t}\leq n}\left|A_{j_{1% }}\cap A_{j_{2}}\cap\cdots\cap A_{j_{t}}\right|.

ⓘ

Symbols:: $\left|\NVar{A}\right|$ : number of elements of a finite set $A$ , $\cap$ : intersection, $\setminus$ : set subtraction, $\cup$ : union, $j$ : nonnegative integer, $n$ : nonnegative integer, $A_{k}$ : subsets and $S$ : set
Permalink:: http://dlmf.nist.gov/26.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.18 and Ch.26

…