Computer Science Coursework
Author: Philip Nelson (see also p.213 of text)
\(g(\xi) = 1 - \frac{v}{2}(3 - 4e^{-i\xi h} + e^{-i\xi 2h}) + \frac{v^2}{2}(1 - 2e^{-i\xi h} + e^{-i\xi 2h})\) \(\implies e^{i\xi h} g(\xi) = e^{i\xi h} - \frac{v}{2}(3e^{i\xi h} - 4 + e^{-i\xi h}) + \frac{v^2}{2}(e^{i\xi h} - 2 + e^{-i\xi h})\) \(= e^{i\xi h} - \frac{v}{2}(3e^{i\xi h} - 4 + e^{-i\xi h}) - v^2 + v^2\cos(\xi h)\) \(= e^{i\xi h} - \frac{v}{2}(2e^{i\xi h} - 4) + v^2(\cos(\xi h) - 1)\) \(= e^{i\xi h}(1 - v) - 2v + v\cos(\xi h) + v^2(\cos(\xi h) - 1)\) \(= e^{i\xi h}(1 - v) - v(2 - \cos(\xi h)) + v^2(\cos(\xi h) - 1)\) \(= (\cos(\xi h) + i\sin(\xi h))(1 - v) - v(2 - \cos(\xi h)) + v^2(\cos(\xi h) - 1)\) \(\implies |g(\xi)|^2 = sin^2(\xi h) + \Big( \cos(\xi h)(1 - v) - v(2 - \cos(\xi h)) + v^2(\cos(\xi h) - 1) \Big)^2\) \(\implies \text{stable for } 0 \leq v \leq 2\)
Last Modification date: 5 May 2018