User:Egm4313.s12.team11.Suh/R6.1

Problem Statement

 * Find the fundamental period of $$\cos (n\omega x) \!$$ and $$\sin (n\omega x) \!$$
 * Show that these functions also have period $$p\!$$.
 * Show that the constants $$a_{0}\!$$ is also a periodic function with period $$p\!$$.

Solution

 * Find the fundamental period, and show that these functions have a period of $$p\!$$.

$$p=2L=2\frac{\pi}{\omega}\!$$

$$L = \frac{\pi}{\omega}\!$$

$$f(x+np) = f(x)\!$$

$$f(x) = cos (n\omega x)\!$$

$$f(x+n\frac{2\pi}{n\omega }) = cos (n\omega (x+\frac{2\pi}{n\omega }))\!$$

$$f(x+\frac{2\pi}{\omega}) = cos (n\omega x+2\pi)\!$$

$$f(x+2L) = cos (n\omega x+2\pi)\!$$

$$p=2L\!$$ is the fundamental period for $$cos (n\omega x)\!$$.

$$f(x+np) = f(x)\!$$

$$f(x) = sin (n\omega x)\!$$

$$f(x+n\frac{2\pi}{n\omega }) = sin (n\omega (x+\frac{2\pi}{n\omega }))\!$$

$$f(x+\frac{2\pi}{\omega }) = sin (n\omega x+2\pi)\!$$

$$f(x+2L) = sin (n\omega x+2\pi)\!$$

$$p=2L\!$$ is the fundamental period for $$sin (n\omega x)\!$$.


 * Show that the constants $$a_{0}\!$$ is also a periodic function with period $$p\!$$.

From Fourier Series, $$a_{0}=\frac{1}{2L}\int_{-L}^{L}f(x)dx\!$$

$$f(x) = \cos (n\omega x)\!$$

$$a_{0}=\frac{\omega}{2\pi}\int_{-\frac{\pi}{\omega}}^{\frac{\pi}{\omega}}cos (n\omega x)dx\!$$

$$a_{0}=\frac{\omega}{2\pi}[\frac{sin (n\omega x)}{n\omega}|_{-\frac{\pi}{\omega}}^{\frac{\pi}{\omega}}]\!$$

$$a_{0}=\frac{\omega}{2\pi}[\frac{2sin (n\pi)}{n\omega}]\!$$

$$a_{0}=0\!$$

$$f(x) = \sin (n\omega x)\!$$

$$a_{0}=\frac{\omega}{2\pi}\int_{-\frac{\pi}{\omega}}^{\frac{\pi}{\omega}}sin (n\omega x)dx\!$$

$$a_{0}=\frac{\omega}{2\pi}[\frac{cos (n\omega x)}{n\omega}|_{-\frac{\pi}{\omega}}^{\frac{\pi}{\omega}}]\!$$

$$a_{0}=\frac{\omega}{2\pi}[\frac{-cos{n\pi}+cos{n\pi}}{n\omega}]\!$$

$$a_{0}=0\!$$