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Find a root for this expression:

  1. Jul 12, 2005 #1
    cos(x)+cos(ix)+cos(x*i^3/2)+cos(x*i^1/2)=0 for x

    I have spent a lot of time finding an analytic root to this equation without success. An analytic root may not exist. I don't know. It is roughly equal to (8facorial)^(1/8)
  2. jcsd
  3. Jul 12, 2005 #2


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    I don't think there is an anaylitical solution and I am not sure that there even is a root.

    x = 0 is closer than your x = 3.764350600, but that's only taken it down to 4 rather than 8.0003108.
  4. Jul 12, 2005 #3
    Hmm..I ust realized for the expression above the root must be complex of the form i^1/4.

    Trig functions of the form cosx+cosy =0 can be solved because of the product formula:


    but is there such a formula for cos(x)+cos(y)+cos(z) ?
  5. Jul 13, 2005 #4


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    I did some 3D Plots in mathematica to get a rough idea where a root might be and then used the FindRoot function and got that 1.4405686011239758 - 3.477840254362339i is approximatly a root.

    I don't think there is any simplification like you want.
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