# csch(x) and tanh(x)...does the "h" mean anything?

• Sep 2nd 2007, 05:50 PM
circuscircus
csch(x) and tanh(x)...does the "h" mean anything?
My teacher gave us a worksheet and sometimes I see csch(...) and tanh(...) and other times without the "h"

Does the h mean anything or is this probably a typo?
• Sep 2nd 2007, 05:58 PM
Krizalid
They're the Hyperbolic Functions, we define as follows

$\displaystyle \sinh=\frac{e^x-e^{-x}}2$ and $\displaystyle \cosh=\frac{e^x+e^{-x}}2$
• Sep 2nd 2007, 06:09 PM
circuscircus
Quote:

Originally Posted by Krizalid
They're the Hyperbolic Functions, we define as follows

$\displaystyle \sinh=\frac{e^x-e^{-x}}2$ and $\displaystyle \cosh=\frac{e^x+e^{-x}}2$

based on that...

is tanh =

$\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}$ ?
• Sep 2nd 2007, 06:23 PM
Jhevon
Quote:

Originally Posted by circuscircus
based on that...

is tanh =

$\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}$ ?

yes, $\displaystyle \tanh x = \frac {e^x - e^{-x}}{e^x + e^{-x}}$ but it is usually just defined as $\displaystyle \frac {\sinh x}{\cosh x}$
• Sep 2nd 2007, 06:35 PM
circuscircus
Quote:

Originally Posted by Jhevon
yes, $\displaystyle \tanh x = \frac {e^x - e^{-x}}{e^x + e^{-x}}$ but it is usually just defined as $\displaystyle \frac {\sinh x}{\cosh x}$

how do you integrate these functions? sinh, cosh and tanh?
• Sep 2nd 2007, 06:36 PM
ThePerfectHacker
Quote:

Originally Posted by circuscircus
how do you integrate these functions? sinh, cosh and tanh?

$\displaystyle \int \sinh x dx = \cosh x + C$

$\displaystyle \int \cosh x dx = \sinh x +C$

$\displaystyle \int \tanh x dx = \int \frac{e^x - e^{-x}}{e^x + e^{-x}} dx$ let $\displaystyle u=e^{x}+e^{-x}$. Continue.
• Sep 3rd 2007, 07:44 AM
Krizalid
Quote:

Originally Posted by circuscircus
how do you integrate these functions? sinh, cosh and tanh?

\displaystyle \begin{aligned}\int\sinh\,dx&=\int\frac{e^x-e^{-x}}2\,dx\\&=\frac12(e^x+e^{-x})+k&\\&=\cosh+\,k,\,k\in\mathbb R\end{aligned}

The same for $\displaystyle \cosh$