Functions of a Complex variable

Question

If tan(Θ+i*Φ)=tanα+i*sec(α)

 

Show that e^2Φ=±cot(α/2) and 2Θ=(n+1/2)*Π+α

Answer 1

Again, similar story to what you had before. The main part is to separate Re and Im pars of tan(Θ+i*Φ). If you do that you get

 

tan(α) = sin(2Θ)/(cos(2Θ)+cosh(2Φ))

sec(α) = sin(2Φ)/(cos(2Θ)+cosh(2Φ))

 

Next some manipulations with expressions you have.

Hint:  e^2Φ is basically coming from sinh(2Φ) and (or) cosh(2Φ). So try it.