I have to minimize the following expression with respect to $z$:
$$\left\| e^{-j\beta z}+\|\Gamma\|e^{j\left(\theta +\beta z\right)} \right\|$$
with $ 0 \leq \|\Gamma\| \leq 1$ and $j = \sqrt{-1}$. This is known as the voltage in a transmition line with $\Gamma$ the coefficient of reflection. Hope you can help me out !
On
The sum achieves a minimum modulus when the two terms are in phase or in opposition, i.e.
$$-\beta z=\theta+k\pi.$$
When this is the case, the modulus reduces to
$$\left\|1+\|\Gamma\|e^{-\beta z}\right\|=\left\|1\pm\|\Gamma\|e^\theta\right\|.$$
Take the sign that achieves the smallest value.
After the fix of the formula,
$$-\beta z=\theta+\beta z+k\pi$$
and the solution is
$$\|1-\|\Gamma\|\|.$$
Figured it out:
If we plot this on the complex plane we will notice that the minimum ocurres at:
$$2\beta z + \theta = \pi$$