| Version 7 |
Version 6 |
|
The hyperbolic functions $\sinh$ ({\em sinus hyperbolicus}) and $\cosh$ ({\em cosinus hyperbolicus}) with arbitrary complex argument $x$ are defined as follows:
|
The hyperbolic functions $\sinh x$ and $\cosh x$ with $x\in\mathbb{C}$ are defined as follows:
|
| \begin{eqnarray*} |
\begin{eqnarray*} |
|
\sinh x&:=&\frac{e^x-e^{-x}}{2},\\
|
\sinh x&:=&\frac{e^x-e^{-x}}{2}\\
|
| \cosh x&:=&\frac{e^x+e^{-x}}{2}. |
\cosh x&:=&\frac{e^x+e^{-x}}{2}. |
| \end{eqnarray*} |
\end{eqnarray*} |
|
One can then also also define the functions $\tanh$ ({\em tangens hyperbolica}) and $\coth$ ({\em cotangens hyperbolica}) in analogy to the definitions of $\tan$ and $\cot$:
|
One can then also define the functions $\tanh x$ and $\coth x$ in analogy to the definitions of $\tan x$ and $\cot x$:
|
| \begin{eqnarray*} |
\begin{eqnarray*} |
|
\tanh x&:=&\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}},\\
|
\tanh x&:=&\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\\
|
| \coth x&:=&\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}. |
\coth x&:=&\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}. |
| \end{eqnarray*} |
\end{eqnarray*} |
| The hyperbolic functions are named in that way because the hyperbola |
The hyperbolic functions are named in that way because the hyperbola |
| $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ |
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ |
| can be written in parametrical form with the equations: |
can be written in parametrical form with the equations: |
| $$x=a\cosh t,\quad y=b\sinh t.$$ |
$$x=a\cosh t,\quad y=b\sinh t.$$ |
| This is because of the equation |
This is because of the equation |
| $$\cosh^2 x-\sinh^2 x=1.$$ |
$$\cosh^2 x-\sinh^2 x=1.$$ |
| There are also addition formulas which are like the ones for trigonometric functions: |
There are also addition formulas which are like the ones for trigonometric functions: |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \sinh (x\pm y)&=&\sinh x\cosh y\pm\cosh x\sinh y\\ |
\sinh (x\pm y)&=&\sinh x\cosh y\pm\cosh x\sinh y\\ |
| \cosh (x\pm y)&=&\cosh x\cosh y\pm\sinh x\sinh y. |
\cosh (x\pm y)&=&\cosh x\cosh y\pm\sinh x\sinh y. |
| \end{eqnarray*} |
\end{eqnarray*} |
| The Taylor series for the hyperbolic functions are: |
The Taylor series for the hyperbolic functions are: |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \sinh x&=&\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}\\ |
\sinh x&=&\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}\\ |
| \cosh x&=&\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}. |
\cosh x&=&\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}. |
| \end{eqnarray*} |
\end{eqnarray*} |
|
There are the following \PMlinkescapetext{connections} between the hyperbolic and the trigonometric functions:
|
Using complex numbers we can use the hyperbolic functions to express the trigonometric functions:
|
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \sin x&=&\frac{\sinh (ix)}{i}\\ |
\sin x&=&\frac{\sinh (ix)}{i}\\ |
| \cos x&=&\cosh (ix). |
\cos x&=&\cosh (ix). |
| \end{eqnarray*} |
\end{eqnarray*} |
