S-plane transfer function contour
There is no input, the transfer function is evaluated at \$s=\sigma+j\omega\$ in the whole complex plane. The angle gets positive because the imaginary part, \$j\omega\$, is positive, and \$\mathrm{atan2}(\Im&gt;0,\Re)&gt;0\$ (when the real part is negative, there's a phase jump). And since a picture is worth a thousand words, here's how the graph evolves for:
$$H(s)=\dfrac{s+2}{s+3}$$

The contour is not closed because I wanted to make the directions more clear. The black trace is the contour of \$s\$, the green is the evaluation of \$H(s)\$, the blue bullet is the pole, and the red circle is the zero. The evaluation starts at \$s=-1.625\$ and continues in clockwise direction (so first it goes negative). The evaluation has its first value at \$H(s)=0.2727...\$, and the direction is counter-clockwise (therefore it, too, goes negative first). It's easy to verify, you don't even need to follow the exact numbers I used. Here's a quick test for some round numbers (same values for the pole and the zero):
$$\begin{align}
&amp;H(-1)&amp;=\;&amp;\dfrac{-1+2}{-1+3}&amp;=\;&amp;0.5 \tag{1} \\
&amp;H(-2-j)&amp;=\;&amp;\dfrac{-2-j+2}{-2-j+3}&amp;=\;&amp;0.5-j0.5 \tag{2} \\
&amp;H(-4)&amp;=\;&amp;\dfrac{-4+2}{-4+3}&amp;=\;&amp;2 \tag{3} \\
&amp;H(-2+j)&amp;=\;&amp;\dfrac{-2-j+2}{-2-j+3}&amp;=\;&amp;0.5+j0.5 \tag{4} \\
\end{align}$$
Which tells you that:

(1) \$\omega_z&lt;\Re{(s)}&lt;0\; ,\;\Im{(s)}=0\quad\Rightarrow\quad\Re{(H(s))}&gt;0\; ,\;\Im{(H(s))}=0\$
(2) \$\Re{(s)}&lt;0\; ,\;\Im{(s)}&lt;0\quad\Rightarrow\quad\Re{(H(s))}&gt;0\; ,\;\Im{(H(s))}&lt;0\$
(3) \$\Re{(s)}&lt;\omega_p&lt;0\; ,\;\Im{(s)}=0\quad\Rightarrow\quad\Re{(H(s))}&gt;0\; ,\;\Im{(H(s))}=0\$
(4) \$\Re{(s)}&lt;0\; ,\;\Im{(s)}&gt;0\quad\Rightarrow\quad\Re{(H(s))}&gt;0\; ,\;\Im{(H(s))}&gt;0\$

What's different from the book is the direction of the evaluation of \$H(s)\$. They're showing that the directions go bith CW, while here they're mirrored, but their maginary values are the same: both positive or negative. Maybe I missed something, I'll look at it tomorrow.