Solve Binomial Identities to Approximate Fresnel Zone Radius

[mezarashi]

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) <--- ?
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[arildno2]

Let's set this in Latex:
$$(D^{2}+r^{2})^{\frac{1}{2}}=D(1+\epsilon)^{\frac{1}{2}}, \epsilon=(\frac{r}{D})^{2}$$
If we now have $\epsilon<<1$, we have, by retaining the second term of the Taylor series about $\epsilon=0$:
$$(1+\epsilon)^{\frac{1}{2}}\approx{1}+\frac{\epsilon}{2}$$

Thus, you have:
$$(D^{2}+r^{2})^{\frac{1}{2}}\approx{D}+\frac{r^{2}}{2D}, \frac{r}{D}<<1$$

 

[mezarashi]

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.

 

[arildno2]

Okay, about Taylor series:
You know that given f(x) differentiable, you may write it as:
$$f(x)=f(0)+\int_{0}^{x}f'(t)dt(1)$$
Now, use partial integration on the integral in the following manner:
$$\int_{0}^{x}f'(t)dt=(t-x)f'(t)\mid_{t=0}^{t=x}-\int_{0}^{x}(t-x)f''(t)dt(2)$$
We also have:
$$(t-x)f'(t)\mid_{t=0}^{t=x}=0*f'(x)+xf'(0)=xf'(0)$$
thus, (2) may be written as:
$$f(x)=f(0)+f'(0)x+\int_{0}^{x}(x-t)f''(t)dt(3)$$
where I have drawn the minus sign underneath the integral sign.
We may now rewrite (3) by noting:
$$\int_{0}^{x}(x-t)f''(t)dt=\frac{x^{2}}{2!}f''(0)+\frac{1}{2!}\int_{0}^{x}(x-t)^{2}f'''(t)dt$$
that is:
$$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+\frac{1}{2!}\int_{0}^{x}(x-t)^{2}f'''(t)dt(4)$$
The emerging series has the form, for an infinitely differentiable function:
$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n}$$
where $n!$ is the factorial 1*2*3...*n (0!=1), and $$f^{(n)}$$ 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".