(Solved):   complex analysis  please  answer  all  questions  up to number 2 ...

 

complex analysis 
please  answer  all  questions  up to number 2

2. Simplify completely: (a) \( \sqrt[5]{1+i \sqrt{3}} \) (b) \( (2+3 i)^{1+i} \) Part 2.: 1. Describe and sketch each of the following sets of complex numbers: (a) \( 1<|2 z-8|<6 \) (b) \( |z-2-i|=3 \) (c) \( \operatorname{Re}(i z+2)>0 \) (d) \( |z-1|^{2}+|z+i|^{2}<2 \). 2. : (a) For \( n \geq 1 \), show that \( 1+z+z^{2}+\cdots+z^{n}=\frac{1-z^{n+1}}{1-z}, z \neq 1 \) (b) Hence, show that \( 1+\cos \theta+\cos 2 \theta+\cdots+\cos n \theta=\frac{1}{2}+\frac{\sin \left(n+\frac{1}{2}\right) \theta}{2 \sin \frac{\theta}{2}} \)