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A Maxima problem

  1. Sep 2, 2009 #1
    Let

    [tex]
    F=z_1\cos\theta_1\cos\theta_2+z_2(e^{i\alpha_1}\sin\theta_1\cos\theta_2+e^{i\alpha_2}\cos\theta_1\sin\theta_2)+z_3e^{i(\alpha_1+\alpha_2)}\sin\theta_1\sin\theta_2
    [/tex]

    where [tex] 0\le\alpha_i\le\pi,~0\le\theta_i\le\pi/2 [/tex] and [tex] z_i[/tex]
    are some fixed complex numbers.

    Then how to find
    [tex]\max_{\theta_i, \alpha_i}|F|[/tex]

    We note that [tex]|F|\le1[/tex].

    Particularly, I want to know if there is any set of constraints like the case of optimization over real variables. [I know definitely that the conditions are [tex] \theta_1=\theta_2; \alpha_1=\alpha_2[/tex]. But I have to establish it. So, how to prove it?]
     
  2. jcsd
  3. Sep 4, 2009 #2
    Some corrections, [tex]|F|\le1[/tex]
    should be [tex]|F|\le N \mbox{ i.e., $|F|$ is bounded}[/tex].

    I was trying to use the observation

    [tex]\max|\sum z_i|=\sum|z_i|[/tex]
    occurs iff [tex]z_i[/tex]s have equal argument. But for my case, since [tex]z_i[/tex]s
    are arbitrary, I can't drive out some common phase to get the maximum as [tex]\sum|z_i|[/tex]. So. for generic [tex]z_i,~~|F|[/tex] should depend on [tex]\alpha_i[/tex]s.

    In light of these observations, my precise questions are:

    1. Can we choose [tex] \theta_1=\theta_2[/tex] to get max|F|?
    2. Can we choose [tex] \alpha_1=\alpha_2[/tex] too ?
    3. Is the conditions 1. and 2. are necessary to get |F|?
    4. How?
     
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