Let $z$ be a complex number with $\Re(z) = \dfrac{1}{7}$ and $|z| = 1$. Evaluate $\dfrac{(1+iz)(1+{\overline {z}}^{2})}{1-i{ {\overline {z}}}}$

Question

Let $z$ be a complex number with $\Re(z) = \dfrac{1}{7}$ and $|z| = 1$. Evaluate $$\dfrac{(1+iz)(1+{\overline {z}}^{2})}{1-i{\displaystyle {\overline {z}}}}$$

https://i.stack.imgur.com/Tkxy3.png

Answer 1

$$\dfrac{(1+iz)(1+{\overline {z}}^{2})}{1-i{\displaystyle {\overline {z}}}} =\dfrac{(1+iz)(1+i{\overline {z}})(1-i{\overline {z}})}{1-i{\displaystyle {\overline {z}}}} $$

$$=(1+iz)(1+i{\overline {z}}) = 1+iz+i{\overline {z}}+i^2z{\overline {z}}$$

$$= 1+i\underbrace{(z+{\overline {z}})}_{2\Re(z)}-1 = {2\over 7}i$$