Convert a complex number into trigonometric form.

Question

Convert the complex number $$\Large e^{\frac{iz}{\bar z+1}}$$ $z\in \mathbb C$, into trigonometric form.

any suggestions please?

Answer 1

$$\frac{iz}{\bar z + 1} = \frac{ix-y}{x+1-iy} = \frac{(ix-y)(x+1+iy)}{(x+1)^2+y^2} = \frac{-yx-y-xy+ix-iy^2+ix^2}{(x+1)^2+y^2} = \frac{-y-2xy+i(x-y^2+x^2)}{(x+1)^2+y^2} = a+ib$$

$$\Rightarrow \exp\left(\frac{iz}{\bar z + 1}\right) = e^{a+ib} = e^a \cdot e^{ib} = e^a (\cos(b)+i\cdot \sin(b))$$

I hope I expanded everything correctly.

Alternatively:

$$\frac{iz}{\bar z + 1} = \frac{iz(z+1)}{\bar z z+z+\bar z + 1} $$ Note that $\bar z z+z+\bar z + 1 \in \mathbb R$ After this step you still should choose another representation for $z$.