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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 :

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  1. TrojanPoem
    • one year ago
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    \[\left| \frac{ 1 }{ Z } + Z \right| = a\]

  2. TrojanPoem
    • one year ago
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    a is not equal to zero

  3. TrojanPoem
    • one year ago
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    It's a complex number , you can assume it's A + Bi

  4. TrojanPoem
    • one year ago
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    It is.

  5. TrojanPoem
    • one year ago
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    No problem :D

  6. phi
    • one year ago
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    the the max and min of what ?

  7. TrojanPoem
    • one year ago
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    of the complex number Z

  8. phi
    • one year ago
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    how do we define the max of a complex number? do you mean the max magnitude ?

  9. TrojanPoem
    • one year ago
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    You can the maximum of A , B ( A+ Bi ) is the complex number.

  10. TrojanPoem
    • one year ago
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    Yeah, the magnitude

  11. phi
    • one year ago
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    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
    • one year ago
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    Hmm, do you mean that e^x ? Never heard of it :(

  13. phi
    • one year ago
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    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
    • one year ago
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    exp(i theta) means \( e^{i \theta}\)

  15. TrojanPoem
    • one year ago
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    Oh, Got it

  16. TrojanPoem
    • one year ago
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    Ok now we ave e^thetai + e^-thetai

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