Tagged Questions

1
vote
2answers
75 views

inverse of laplace transform

How to compute this inverse Laplace transform ? $$\displaystyle{ \mathcal{L^{-1}} \left\{ \frac{1}{s(\exp(s)+1)} \right\} }$$ Thanks.
5
votes
2answers
99 views

Help finding inverse of $f(x)=\frac{e^x-e^{-x}}{2}$

I'm trying to find the inverse of $f(x)=\frac{e^x-e^{-x}}{2}$. My textbook says $f^{-1}(x)=\ln(x+\sqrt{x^2+1})$, but I haven't been able to get that answer. Switching $x$ and $y$, I tried solving for ...
2
votes
3answers
264 views

The relation between an exponential function and a logarithmic function

I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ...
4
votes
2answers
128 views

Solve equation $\tfrac 1x (e^x-1) = \alpha$

I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for ...
4
votes
2answers
105 views

$\ln(x)$, $e^{x}$ and $\int \frac{1}{x}dx$ relationship

My math professor told me that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$ by the definition; so far so good. But how/why does $\ln(x)$ ($\int_1^x\frac{1}{t} dt$: by defintion) coincide with the inverse of ...