Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Staff Emeritus
    Science Advisor

    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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Complex numbers
  1. Complex numbers (Replies: 15)

  2. Complex number (Replies: 6)

  3. Complex numbers? (Replies: 6)

  4. Complex numbers (Replies: 2)

  5. Complex numbers (Replies: 2)

Loading...