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

1. 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?

2. They're the Hyperbolic Functions, we define as follows

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

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

$\sinh=\frac{e^x-e^{-x}}2$ and $\cosh=\frac{e^x+e^{-x}}2$
based on that...

is tanh =

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

4. Originally Posted by circuscircus
based on that...

is tanh =

$\frac{e^x-e^{-x}}{e^x+e^{-x}}$ ?
yes, $\tanh x = \frac {e^x - e^{-x}}{e^x + e^{-x}}$ but it is usually just defined as $\frac {\sinh x}{\cosh x}$

5. Originally Posted by Jhevon
yes, $\tanh x = \frac {e^x - e^{-x}}{e^x + e^{-x}}$ but it is usually just defined as $\frac {\sinh x}{\cosh x}$
how do you integrate these functions? sinh, cosh and tanh?

6. Originally Posted by circuscircus
how do you integrate these functions? sinh, cosh and tanh?
$\int \sinh x dx = \cosh x + C$

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

$\int \tanh x dx = \int \frac{e^x - e^{-x}}{e^x + e^{-x}} dx$ let $u=e^{x}+e^{-x}$. Continue.

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