Chapter 14

# 14.5 Concept of Mapping

Consider a map (function) $F(s)$:

$F(s)=$ $\frac{s+5}{(s+1)(s+3)}$

The map (function) has $3$ singularities: a zero at $-5$, and two poles, at $-1$ and $-3$.

Consider an arbitrarily closed contour $\sigma$ in the s-plane, traversed clockwise (CW), so that it does not go through any singularities of $F(s)$. Let $Z$ be a number of zeros of $F(s)$ inside the $\sigma$-contour, and $P$ be a number of poles of $F(s)$ inside the $\sigma$ -contour. Mapping the $\sigma$ -contour into the $F(s)$ plane will result in a closed $\Gamma$ -contour. Let $N$ be a number of clockwise (CW) encirclements of the resulting $\Gamma$-contour around the origin of the $F(s)$-plane.

14.5.1 Case 1

Let the $\sigma$-contour be so chosen that Z = 0 and P = 0, i.e. there are no singularities of $F(s)$ inside the $\sigma$ – contour. Note that the resulting $\Gamma$ -contour in the $F(s)$-plane does not encircle the origin of $F(s)$-plane, i.e. N = 0.

14.5.2 Case 2

Next, let the $\sigma$ -contour be so chosen that $Z =1$ and P = $0$, i.e. there is one zero of $F(s)$ inside the $\sigma$ -contour. Note that the resulting $\sigma$-contour in the $F(s)$-plane encircles the origin of $F(s)$-plane once in a clockwise (CW) direction, i.e. N =$+1$.

14.5.3 Case 3

Next, let the $\sigma$-contour be so chosen that $Z =0$ and $P = 1$, i.e. there is one pole of $F(s)$ inside the $\sigma$-contour. Note that the resulting $\Gamma$ -contour in the $F(s)$-plane encircles the origin of $F(s)$-plane once in a counter-clockwise (CCW) direction, i.e. $N = -1$.

14.5.4 Case 4

Next, let the $\sigma$-contour be so chosen that $Z = 1$ and $P = 1$, i.e. there is one zero and one pole of $F(s)$ inside the $\sigma$ -contour. Note that the resulting $\Gamma$-contour in the $F(s)$-plane does not encircle the origin of $F(s)$-plane, i.e. $N = 0.$

14.5.5 Case 5

Next, let the $\sigma$-contour be so chosen that $Z = 0$ and $P = 2$, i.e. there are two poles of $F(s)$ inside the $\sigma$-contour. Note that the resulting $\Gamma$-contour in the $F(s)$-plane encircles the origin of $F(s)$-plane twice in a counter-clockwise (CCW) direction, i.e. $N = -2.$

14.5.6 Case 6

Next, let the $\sigma$-contour be so chosen that $Z = 1$ and $P = 2$, i.e. there is $1$  zero and $2$ poles of $F(s)$ inside the $\sigma$ -contour. The area of the origin may not be very visible, so see a zoomed version in Error! Reference source not found. Note that the resulting $\Gamma$-contour in the $F(s)$-plane encircles the origin of $F(s)$-plane once in a counter-clockwise (CCW) direction, i.e. $N = -1$.