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Approximation with small parameter

  1. Jan 19, 2016 #1
    For some small parameter ##\epsilon##, how would one go about making an approximation such as ##\sqrt{k^2-\epsilon^2}\approx k-\frac{\epsilon^2}{2k}##? I was thinking that these types of approximations came from truncating Taylor series expansions, but I can't see how it would be obvious which parameter one would differentiate with respect to, a priori, or what value to expand about. Could someone please explain the general method for making such approximations? Thanks!
  2. jcsd
  3. Jan 19, 2016 #2


    Staff: Mentor

    More specifically, a binomial series expansion, writing ##\sqrt{k^2-\epsilon^2}## as ##(k^2 - \epsilon^2)^{1/2}##. That's where I would start.
  4. Jan 19, 2016 #3
    You can use the binomial approximation by first pulling the k term out.

    √(k^2 - e^2) = k⋅√(1 - (e/k)^2) = k⋅(1 - (e/k)^2)^(1/2)

    Now look at the binomial approximation.
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