Why is the Z-transform of $e^{at}$, t = kT, different from Laplace transform of $e^{at}$

Question

The Laplace transform of $e^{at}$ takes a well known form of $\frac{1}{s-a}$

The Z transform of $e^{at} = e^{akT} $ T is the sampling period takes the form of $\frac{z}{z-e^{aT}}$

Does anyone know why the Z-transform takes a different form compared to the laplace transform despite both z and s being complex numbers?

Can we get from one to the other and how?

Thanks

Answer 1

$Z$-transform is applied to discrete functions, whereas Laplace transform to non-discrete ones.

In your example, you compute actually the $Z$-transform of $e^{ak}$, where $k$ is an integer representing the discretization of $t$ in a period $T$. Thus, $e^{ak}$ is a discretization of $e^{at}$, but not the same function.