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Aproximations by Expansion

  1. Jul 14, 2006 #1

    eep

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    Hi,
    I've never really studied various ways of expanding expressions in order to obtain an approximation that can make calculations easier. For example,

    [tex]
    p^t = \frac{m}{\sqrt{1 - v^2}}
    [/tex]

    reduces to

    [tex]
    p^t = m + \frac{1}{2}mv^2 + ...
    [/tex]

    for v << 1.

    How does one arrive at something like this? What other expansions are useful? I used to think that I'd just calculate everything exactly but I now realize these sorts of expansions are extremeley important.
     
  2. jcsd
  3. Jul 14, 2006 #2

    quasar987

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    Are you familiar with Taylor series? It says that if a function can be expanded in a power series, then it will be of the form

    [tex]f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n[/tex]

    Wheter this series converges towards the original function is another matter. (Actually, it is a matter of calculating the radius of convergence of a power series)

    For exemple, the function [itex]f(x) = (1+x)^r[/itex] where r is any real number has as its Taylor expansion

    [tex]\sum_{n=0}^{\infty} \frac{r(r-1)...(r-n+1)}{n!}x^n[/tex]

    and the series converges to [itex]f(x) = (1+x)^r[/itex] for all |x|<1 but not for any other value of x.

    This is exactly what has been done in your post. [itex]p(v)=m(1+v^2)^{-1/2}[/itex] is of the form [itex](1+x)^r[/itex] with x=-v² and r=-½, so it converges to the series expansion you wrote for all values of v such that |v²|<1.

    N.B. in the context of special relativity, v is always lesser than 1 since it has been assumed that c=1, so the formula really is valid for all velocity.


    The only other expansion I know of is the expansion in a Fourier series.
     
    Last edited: Jul 15, 2006
  4. Jul 14, 2006 #3

    quasar987

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    Btw, wouldn't it be
    [tex]
    p = m - \frac{1}{2}mv^2 + ...
    [/tex]
    instead? (with a "-" sign instead of a "+" sign on odd terms)
     
  5. Jul 15, 2006 #4

    mathman

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    The sign is +. It results from multiplying two - signs, -1/2 and -v2.
     
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