Theory

The complex variables z and s are related by the equation:
$$ z = e^{Ts} \tag{1} $$

This means that 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 $$ s = \sigma + j \omega \tag{2} $$

and

$$ z = e^{T(\sigma +j \omega)} = e^{T \sigma} e^{jT \omega} = e^{T \sigma} e^{j(T \omega + 2 \pi k)} \tag{3} $$

From this last equation we see that poles and zeros in the s plane, where frequencies differ in intefral 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

$$ |z| = e^{T \sigma} \lt 1 \tag{4} $$

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.