## TrojanPoem one year ago Z is a complex number, a is a constant. Find the maximum and minimum value of the complex number Z notice that :

1. TrojanPoem

$\left| \frac{ 1 }{ Z } + Z \right| = a$

2. TrojanPoem

a is not equal to zero

3. TrojanPoem

It's a complex number , you can assume it's A + Bi

4. TrojanPoem

It is.

5. TrojanPoem

No problem :D

6. phi

the the max and min of what ?

7. TrojanPoem

of the complex number Z

8. phi

how do we define the max of a complex number? do you mean the max magnitude ?

9. TrojanPoem

You can the maximum of A , B ( A+ Bi ) is the complex number.

10. TrojanPoem

Yeah, the magnitude

11. phi

you could write Z= a exp( i x) so Z+1/Z = a exp(ix) + (1/a) exp(-ix) and the magnitude squared is | Z+1/Z |^2 = (a exp(ix) + (1/a) exp(-ix))(a exp(-ix) + (1/a) exp(ix)) which might lead to a solution

12. TrojanPoem

Hmm, do you mean that e^x ? Never heard of it :(

13. phi

no. I meant a complex number in polar coords is magnitude * exp( i* theta) btw, I should have used r instead of a, to not confuse the magnitude with the "a" given in your problem.

14. phi

exp(i theta) means $$e^{i \theta}$$

15. TrojanPoem

Oh, Got it

16. TrojanPoem

Ok now we ave e^thetai + e^-thetai