We all know that $a^nu(n)$ has unilateral $Z$-transform. But what is the $Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'.

But bilateral Laplace transform of $e^t$ doesn't exist. Both are exponentials in discrete and continuous domain respectively. Considering the similarity between Laplace and $Z$-transform, how to explain the above problem?

Below, this is how I got 'zero'

$a^n$=$a^nu(n)$ + $a^nu(-n-1)$ ,Now taking Z-transform on both sides we get $z/(z-a)$ and $-z/(z-a)$ respectively which add to 'zero'

We all know that $a^nu(n)$ has unilateral $\mathcal Z$-transform. But what is the $\mathcal Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'.

But bilateral Laplace transform of $e^t$ doesn't exist. Both are exponentials in discrete and continuous domain respectively. Considering the similarity between Laplace and $\mathcal Z$-transform, how to explain the above problem?

Below, this is how I got 'zero'

$$a^n=a^nu(n) + a^nu(-n-1),$$

Now taking $\mathcal Z$-transform on both sides we get $$\frac{z}{z-a}\quad \text{and}\quad \frac{-z}{z-a}$$ respectively which add to 'zero'

We all know that $a^nu(n)$ has unilateral $Z$-transform. But what is the $Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'.

But bilateral Laplace transform of $e^t$ doesn't exist. Both are exponentials in discrete and continuous domain respectively. Considering the similarity between Laplace and $Z$-transform, how to explain the above problem?

We all know that $a^nu(n)$ has unilateral $Z$-transform. But what is the $Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'.

But bilateral Laplace transform of $e^t$ doesn't exist. Both are exponentials in discrete and continuous domain respectively. Considering the similarity between Laplace and $Z$-transform, how to explain the above problem?

Below, this is how I got 'zero'

$a^n$=$a^nu(n)$ + $a^nu(-n-1)$ ,Now taking Z-transform on both sides we get $z/(z-a)$ and $-z/(z-a)$ respectively which add to 'zero'