Functions of Complex Variable

Question

(a) If cos(Θ+i*Φ)=R(cosα+i*sinα),prove that e^2Φ=sin(Θ-α)/sin(Θ+α)

 

(b) if u=log(tan(Π/4+Θ/2)),prove that Θ=-i*log(tan( (Π/4) + (i*u)/2))

Answer 1

Hello Ken,

(a)

We know that

cos(Θ+i*Φ)=
     = cos(Θ) cosh(Φ) -i sin(Θ) sinh(Φ)=R[cos(α)+i sin(α)]
Rcosα=cos(Θ)cosh(Φ)   (1) (multiply by sin(Θ)

Rsinα=sin(Θ)sinh(Φ)    (2)  (multiply by cos(Θ)

 

Take sum (1) and (2) then their differences and then divide them.

 

You will have

[sin(Θ+α)/sin(Θ-α)=[cosh(Φ)+sinh(Φ)]/[cosh(Φ)-sinh(Φ)]=e^2Φ

So, (a) is done!

If any questions, let me know