- #1

James Brady

- 105

- 4

Taking the first one, we can manipulate the square root algebraically ##z_0\sqrt{1 + (x-s/2)^2/z_0^2}##, this is where I run into issues, according to the text we can use the fact that ##\sqrt{1+a} = 1 + a/2## in a power series expansion to get ##z_0 + (x-s/2)^2/2 z_0## for small values of x.

I don't know how they got that. When I do a Taylor series expansion on ##z_0\sqrt{1 + (x-s/2)^2/z_0^2}## I get ##z_0\sqrt{1 + (s/2)^2/z_0^2} - \frac{-s/2}{z_0*\sqrt{\frac{-2/2}{z_0^2}+1}} *x## ...

How did they come up with such a simple answer using ##\sum_{n=0}^1 \frac{f^n(0)}{n!} x^n## ?

Am I using the Taylor series wrong?

The book goes on to use ##2cos(\theta) = exp(i \theta) + exp(-i \theta)## to obtain a final wave equation (at the screen) of ##exp(i \theta) cos(ksx/2z_0)## where k is the wave number ##\lambda = 2 \pi/k##

How did they come up with that? I don't see the algebra at all.

Sincerely, confused.