What type of expansion is this?

  1. When I am reading the paper about Rayleigh instability, I found this type of expanding method.

    [tex]
    \sqrt{1+(\frac{2\pi\delta}{\lambda})^2 \cos^2(\frac{2\pi x}{\lambda})} = 1 + \frac{1}{2}(\frac{2\pi\delta}{\lambda})^2\cos^2 (\frac{2\pi x}{\lambda}) + \cdots
    [/tex]

    Can someone tell me what type of expansion is this?
     
  2. jcsd
  3. AlephZero

    AlephZero 7,298
    Science Advisor
    Homework Helper

    It's the Binomial series expansion of ##(1+x)^{1/2}##
     
  4. If we suppose that ##f(x) = (\frac{2\pi\delta}{\lambda})^2 \cos^2(\frac{2\pi x}{\lambda})##, is it right that we could take derivative with respect to ##f(x)## to get the Taylor Expansion? The first order derivative I mean is presented below:

    [tex]
    \frac{d(1 + f(x))^{\frac{1}{2}}}{df(x)} = \frac{1}{2} \frac{1}{\sqrt{1 + f(x)}}
    [/tex]
     
  5. AlephZero

    AlephZero 7,298
    Science Advisor
    Homework Helper

    I'm not sure where that is leading to. I meant
    ##(1+x)^n = 1 + n x + n(n-1)x^2 / 2! + n(n-1)(n-2)x^3/3! + \dots##
    where x is the trig function and n = 1/2.
     
  6. OK. Now I comprehend. Thanks a lot.
     
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