Can someone explain complex mapping as appears in Ogata's textbook
If we denote $s=\sigma+j\omega$, then for the mapping $z=e^{sT}$, the negative real axis in the $z$-plane in the range $z\in[0,-1]$ corresponds to $\omega=\pm\omega_s/2$ (i.e. Nyquist) and $\sigma\in [-\infty,0]$, where $\omega_s$ is the sampling frequency in radians. So if the curve in the $s$-plane moved along the lines $s=\pm j\omega_s/2$ then the curve in the $z$-plane would move on the negative real axis of the $z$-plane. If it doesn't (as suggested in the figure) then this range of the $z$-plane will be excluded. So if with this mapping you want poles on the negative real axis of the $z$-plane, you need corresponding poles with imaginary parts $\pm \omega_s/2$ in the $s$-plane.