Hyperbolic cosine

The hyperbolic cosine is, in Mathématiques, a hyperbolic Fonction.

Definition

The hyperbolic function cosine, noted cosh (sometimes, but more rarely, CH ) is the complex function following:

where $e$ is the complex function Exponentielle.

The hyperbolic function cosine is to some extent the analog of the function Cosinus in the hyperbolic Géométrie.

Properties

General properties

• cosh is continuous and infinitely derivable.
• the derivative of cosh is sinh, the hyperbolic function Sinus.
• the Primitive of cosh is sinh+C, except for a constant of integration C.
• the restriction of cosh on $\backslash \; mathbb\; R$ is even and strictly increasing on $\backslash \; mathbb\; R^+$.

Trigonometrical properties

From share the definitions of the function cosine and hyperbolic sine, one can deduce the following equalities:

$e^z\; =\; \backslash \; cosh\; (Z)\; +\; \backslash \; sinh\; (Z)$

$e^\; \{-\; Z\}\; =\; \backslash \; cosh\; (Z)\; -\; \backslash \; sinh\; (Z)$

These equalities are similar to the Formule of Euler in traditional trigonometry.

Just as the coordinates (cos (T), sin (T)) define a Cercle, (cosh (T), sinh (T)) define the positive branch of a equilateral hyperbole. One has indeed for t>0:

$\backslash \; cosh^2\; (T)\; -\; \backslash \; sinh^2\; (T)\; =\; 1$.

In addition, for $x\; \backslash \; in\; \backslash \; mathbb\; R$:

$\backslash \; cosh\; (I\; X)\; =\; \backslash \; frac\; \{(e^\; \{I\; X\}\; +\; e^\; \{-\; I\; X\})\}\{2\}\; =\; \backslash \; cos\; (X)$

$\backslash \; cosh\; (X)\; =\; \backslash \; cos\; (I\; X)$

$\backslash \; cosh\; (x+y)\; =\; \backslash \; cosh\; (X)\; \backslash \; cosh\; (there)\; +\; \backslash \; sinh\; (X)\; \backslash \; sinh\; (there)$

$\backslash \; cosh^2\; \backslash \; left\; (\backslash \; frac\; \{X\}\; \{2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{1+\; \backslash \; cosh\; (X)\}\{2\}$

Development in Taylor series

cosh, being indefinitely derivable, has a development in Taylor series in any point:

$\backslash \; cosh\; Z\; =\; 1\; +\; \backslash \; frac\; \{z^2\}\; \{2!\}\; +\; \backslash \; frac\; \{z^4\}\; \{4!\}\; +\; \backslash \; frac\; \{z^6\}\; \{6!\}\; +\; \backslash \; cdots\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{z^\; \{2n\}\}\; \{(2n)!\}$

Values

Some values of cosh:

• $\backslash \; cosh\; (0)\; =\; 1$
• $\backslash \; cosh\; (1)\; =\; \backslash \; frac\; \{e^2+1\}\; \{2nd\}$
• $\backslash \; cosh\; (I)\; =\; \backslash \; cos\; (1)$

Reciprocal function

cosh admits a reciprocal Fonction, noted arccosh (or arcosh). It is about a Fonction to multiple values complex. Its connects principal is generally selected by posing like cut the segment $\backslash \; left]\; -\; \backslash \; infty;\; 1\; \backslash \; right$

$\backslash \; operatorname\; \{arcosh\}\; (Z)\; =\; \backslash \; ln\; (Z\; +\; \backslash \; sqrt\; \{z+1\}\; \backslash \; sqrt\; \{z-1\})$

For $x\; \backslash \; in\; \backslash \; left\; the\; restriction\; of\; cosh\; on$ \backslash \; mathbb\; R$admits\; two\; reciprocal:$ arcosh\; (X)\; =ln\; \backslash \; left\; (X\; \backslash \; pm\; \backslash \; sqrt\; \{x^2-1\}\; \backslash \; right)$.$

Physics

The curve representative of the function cosh on $\backslash \; mathbb\; R$ describes a Chaînette, i.e. the shape of a cable fixed at the two ends and subjected to gravity.

See too

• hyperbolic Function

Random links:

Armand Emmanuel of Plessis de Richelieu | Jacques Marette | András Zicsi | Fabrice Josso | Giovanni Maria Bononcini | Peekskill_(station_de_Métro-Nord)

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org