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Complex numbers

  1. Aug 9, 2012 #1
    1- is there any complex number, x ,such that x^x=i?

    2- (-1)^([itex]\sqrt{2}[/itex])=?
     
  2. jcsd
  3. Aug 9, 2012 #2

    HallsofIvy

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    Yes, but finding it is non-trivial, involving, I think, the Lambert W function.

    We can write -1 in "polar form" as [itex]e^{i\pi}[/itex] and then [itex](-1)^{\sqrt{2}}= (e^{i\pi})^{\sqrt{2}}= e^{i\pi\sqrt{2}}= cos(\pi\sqrt{2})+ i sin(\pi\sqrt{2})[/itex]
    or about .99+ .077i.
     
  4. Aug 9, 2012 #3
    Thanks.
     
  5. Aug 15, 2012 #4

    haruspex

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    Writing z = re, zz = i gives θ sec(θ) eθ tan(θ) = π/2 + 2πn and r = eθ tan(θ). For n = 0, θ has a solution in (π/6, π/4), and probably infinitely many for each n.
     
    Last edited: Aug 15, 2012
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