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Chebyshev cubic roots

  1. Jul 10, 2009 #1
    Say I have a monic polynomial,

    x^3 + ax^2 + bx + c

    with a=-2.372282, b=1.862273, c=-0.483023

    The discriminant is given by

    a^2 b^2 - 4 b^3 - 4 a^3 c - 27 c^2 + 18 ab c

    which is < 0, indicating 1 real root and 2 complex conjugates.

    A method for solving a general cubic using the Chebyshev root is explained here,

    but (a^3 - 3b) is negative, which means that "t" will be imaginary. But t is then used in the Chebyshev cubic root, which is only defined for real numbers [-2, inf].
  2. jcsd
  3. Jul 10, 2009 #2


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    The site you give does NOT say that the Chebyshev cubic root is "only defined for real numbers [-2, inf]". It gives simple formulas for the root in the case that the argument is in that interval, and then tells how to expand it to complex arguments.
  4. Jul 10, 2009 #3
    EDIT - deleted previous response.

    Let me clarify my remaining confusion from here:

    They define s = t^2, then say

    Well, as they already pointed out t is imaginary, but then they say [tex]S_{1\over3}(t)[/tex] is a function of a real variable! But it's still defined in terms of t only! So how can I calculate [tex]S_{1\over3}(t)[/tex] given that I can't compute the imaginary t?
    Last edited: Jul 11, 2009
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