Theory:

The complex variables z and s are related by the equation:
$$ z = e^{Ts} \tag{1} $$ This means thtat a pole in the s plane can be located in the z plane through the transformation. Since the complex variable s had real part σ and imaginary part ω, we have $$ s = σ + j ω \tag{2} $$ and $$ z = e^{T(σ +j ω)} = e^{T σ} e^{jT ω} = e^{T σ} e^{j(T ω + 2 πk)} \tag{3} $$ From this last equation we see that poles and zeros in the s plane, where frequencies differ in integral multiples of the sampling frequency 2 π/T , are mapped into the same location in the z plane. This means that there are infinitely many values of s for each value of z.

Since σ is negative in the left half of the s plane, the left half of the s plane corresponds to
$$ |z| = e^{T σ} \lt 1 \tag{4} $$ The j ω axis in the s plane corresponding to |z| = 1. That is, the imaginary axis in the s plane (the line σ = 0) corresponding to the unit circle in the z plane, and the interior of the unit circle corresponds to the left half of the s plane.