Finding Γ given complex number ratio

Question

Originally, I have (1+Γ) / ($e^{-j\beta z} + Γ*e^{j\beta z}$) = 3/2. I understand I must get rid of the complex number on the denominator, but I do not know how to do this. Please help with this simplification at least.

Answer 1

Rearrange the equation

$$1+Γ = \frac32e^{-j\beta z} + \frac32Γe^{j\beta z}$$

Solve for Γ

$$Γ = \frac{1-\frac32e^{-j\beta z}}{\frac32e^{j\beta z}-1} =\frac{(1-\frac32e^{-j\beta z})(\frac32e^{-j\beta z}-1)}{(\frac32e^{j\beta z}-1)(\frac32e^{-j\beta z}-1)} =-\frac{(\frac32e^{-j\beta z}-1)^2}{\frac94+1}=-\frac{1}{13}(3e^{-j\beta z}-2)^2$$