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Help simplifying this equation

  1. Nov 30, 2009 #1
    I'm trying to follow some notes in class.

    so the original equation is:
    u= [ (x^2 – 1)^1/2 + ln (x + (x^2 – 1)^1/2 )]

    and we let x=z+1.

    Then by simplifying and approximating, u=z*2*2^.5 + higher order terms

    from my notes: (x^2 – 1)^1/2 = (2z)^.5*(1+z/2)^.5
    ln(1+e) approximately = e
  2. jcsd
  3. Dec 1, 2009 #2


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    I presume those are the two lines that have you confused?

    The first one is most easily seen by noting that
    x2 - 1 = (x + 1)(x - 1)
    which, if z = x - 1, is z(z + 2). Then the algebra is straightforward (take out a factor of 2 and split the root of product into product of roots).

    For the second one, you can make a Taylor series expansion around x = 0 of
    ln(1 + x) = ln(1) + 1/(1 + 0) x + O(x2)
    and see that to first order, you get x.
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