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Binomial Identities?

  1. Oct 7, 2005 #1

    mezarashi

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    Homework Helper

    I was going the derivations for Fresnel zone radius approximation, and there was a jump in the math which I don't fully understand. If someone could take a look at this and help me figure. I was hinted that it had to do with the binomial theorem, but I have no idea >.<

    Seems like LaTeX isn't working -_-
    (D^2 + r^2)^(1/2)
    if r << D
    ( D^2( 1 + (r^2)/(D^2) ) )^(1/2)

    D1 + (r^2)/(2D^2) <--- ???
    .
    .
    .
     
  2. jcsd
  3. Oct 7, 2005 #2
    Let's set this in Latex:
    [tex](D^{2}+r^{2})^{\frac{1}{2}}=D(1+\epsilon)^{\frac{1}{2}}, \epsilon=(\frac{r}{D})^{2}[/tex]
    If we now have [itex]\epsilon<<1[/itex], we have, by retaining the second term of the Taylor series about [itex]\epsilon=0[/itex]:
    [tex](1+\epsilon)^{\frac{1}{2}}\approx{1}+\frac{\epsilon}{2}[/tex]

    Thus, you have:
    [tex](D^{2}+r^{2})^{\frac{1}{2}}\approx{D}+\frac{r^{2}}{2D}, \frac{r}{D}<<1[/tex]
     
    Last edited: Oct 7, 2005
  4. Oct 7, 2005 #3

    mezarashi

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    Hey awesome, thanks. Oh and Latex is working :P
    Could you tell me a bit more about this "Taylor series" approximation or a link to its derivation, although I know this is true by punching some test numbers in the calculator. Thanks again.
     
    Last edited: Oct 7, 2005
  5. Oct 7, 2005 #4
    Okay, about Taylor series:
    You know that given f(x) differentiable, you may write it as:
    [tex]f(x)=f(0)+\int_{0}^{x}f'(t)dt(1)[/tex]
    Now, use partial integration on the integral in the following manner:
    [tex]\int_{0}^{x}f'(t)dt=(t-x)f'(t)\mid_{t=0}^{t=x}-\int_{0}^{x}(t-x)f''(t)dt(2)[/tex]
    We also have:
    [tex](t-x)f'(t)\mid_{t=0}^{t=x}=0*f'(x)+xf'(0)=xf'(0)[/tex]
    thus, (2) may be written as:
    [tex]f(x)=f(0)+f'(0)x+\int_{0}^{x}(x-t)f''(t)dt(3)[/tex]
    where I have drawn the minus sign underneath the integral sign.
    We may now rewrite (3) by noting:
    [tex]\int_{0}^{x}(x-t)f''(t)dt=\frac{x^{2}}{2!}f''(0)+\frac{1}{2!}\int_{0}^{x}(x-t)^{2}f'''(t)dt[/tex]
    that is:
    [tex]f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+\frac{1}{2!}\int_{0}^{x}(x-t)^{2}f'''(t)dt(4)[/tex]
    The emerging series has the form, for an infinitely differentiable function:
    [tex]f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n}[/tex]
    where [itex]n![/itex] is the factorial 1*2*3...*n (0!=1), and [tex]f^{(n)}[/tex] denotes the n'th derivative of f. (f is considered its own 0'th derivative).

    That infinite series is called the Taylor series of f with respect to 0, a finite, truncated version of it is called a Taylor series approximation to f.

    If we have the identity containing f(x) on the left-hand side and a finite sum and the integral on the right-hand side (for example our (4)), we call the integral "the remainder", or "error term".
     
    Last edited: Oct 7, 2005
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