User:Egm6341.s10.team3.sa/Mtg47

Mtg 47: Mon, 21 Jun 10

 Typeset of transparencies, not lecture transcript. Subramanian Annamalai 17:43, 14 August 2010 (UTC)

[[media: Egm6341.s10.mtg47.djvu | Page 47-1]]  From (Eq.2) on [[media: Egm6341.s10.mtg46.djvu | Page 46-5]],

$$\displaystyle a_k = \frac{2}{n} \underset{i=0}{\overset{n}{{\sum}^{\prime\prime}}} z_k(\theta_i) $$ $$\displaystyle (Eq. 1) $$

$$\displaystyle z_k(\theta_i):= f(cos \theta_i) cos k \theta_i$$ $$\displaystyle (Eq. 2) $$

$$\displaystyle \theta_i := \frac{i \pi}{n}$$ $$\displaystyle (Eq. 3) $$

$$\displaystyle (Eq.1) \Rightarrow a_k = \frac{2}{n}\bigg[\frac{f(1)}{2} + \sum_{i=1}^{n-1}z_k(\theta_i)+ \frac{f(-1)}{2} (-1)^k \bigg]$$ $$\displaystyle (Eq. 4) $$

$$\displaystyle \frac{f(1)}{2} \rightarrow i=0 \Rightarrow \theta_0 = 0 \Rightarrow cos \theta_0 = 1 \Rightarrow cos k\theta_0 = 1 $$

$$\displaystyle \frac{f(-1)}{2} \rightarrow i=n \Rightarrow \theta_n = \pi \Rightarrow cos \theta_n = -1 \Rightarrow cos k\theta_n = (-1)^k $$

(Eq.6) on [[media: Egm6341.s10.mtg46.djvu | Page 46-3]]: Need only $$\displaystyle a_{2k},\; k=0,1, \cdots, \infty$$

$$\displaystyle (Eq.4) \Rightarrow a_{2k} = \frac{2}{n}\bigg[\frac{f(1)}{2} + \underbrace{\sum_{i=1}^{n-1}z_{2k}(\theta_i)}_{f(cos \theta_i)cos2k\theta_i}+ \frac{f(-1)}{2} (-1)^k \bigg]$$

$$\displaystyle cos \alpha = -cos (\pi - \alpha)$$ $$\displaystyle cos \theta_1 = -cos (\underbrace{\pi - \theta_1}_{\theta_{n-1}})$$   $$\displaystyle cos 2k\theta_{n-1} = cos 2k(pi - \theta_1) = cos (2k\pi - 2k\theta_1)$$ 

[[media: Egm6341.s10.mtg47.djvu | Page 47-2]] 

$$\displaystyle cos(2k \theta_{(n-1)}) = cos 2k\theta_1$$



$$\displaystyle (Eq.4) \Rightarrow a_{2k} = \frac{2}{n}\bigg[\frac{f(1)}{2} + \underbrace{\sum_{i=1}^{\frac{n}{2}-1}z_{2k}(\theta_i) + z_{2k}(\theta_{n-i})}_{\big[f(cos \theta_i) + f(-cos \theta_i)\big]cos2k\theta_i}+ \underbrace{f(0)(-1)^k}_{cos\theta_{\frac{n}{2}} = cos \frac{\pi}{2}= 0\,;\, cos2k\theta_{\frac{n}{2}} = cos k\pi} + \frac{f(-1)}{2} \bigg]$$

[[media: Egm6341.s10.mtg47.djvu | Page 47-3]] 

Weights: (Eq.1),(Eq.6) on [[media: Egm6341.s10.mtg46.djvu | Page 46-3]] Goal: Find weights $$\displaystyle w_i, \, i=0,1,2 \cdots n$$cosine series of $$\displaystyle f(cos \theta) $$

(Eq.5) on [[media: Egm6341.s10.mtg46.djvu | Page 46-3]] $$\displaystyle \Rightarrow f(cos \theta) = \underbrace{\bar{a}_0}_{\frac{a_0}{2}} + \sum_{k=1}^{\infty} \underbrace{\bar{a}_k}_{a_k} cosk\theta $$ 

$$\displaystyle \Rightarrow f(cos \theta) = \sum_{k=1}^{\infty} \bar{a}_k cosk\theta $$ $$\displaystyle (Eq. 1) $$

$$\displaystyle \bar{a}_0 = \frac{a_0}{2} = \frac{1}{\pi} \int_{0}^{\pi} f(cos \theta) \,d\theta \;\;\; and \;\; \bar{a}_k = a_k \rightarrow $$ (Eq.7) on [[media: Egm6341.s10.mtg46.djvu | Page 46-3]]

$$\displaystyle \bar{a}_k = \frac{d_k}{\pi} \int_{0}^{\pi} \underbrace{f(cos\theta)cosk\theta}_{z_k(\theta)} \,d\theta $$ where,

$$\displaystyle \begin{align} d_k &= 1 \;for \, k=0 \\ &= 2 \;for \, k>0 \; i.e,\,k=1,2,\cdots \infty \end{align}$$ $$\displaystyle (Eq. 2) $$

[[media: Egm6341.s10.mtg47.djvu | Page 47-4]] 

(Eq.6) on [[media: Egm6341.s10.mtg46.djvu | Page 46-3]] $$\displaystyle \Rightarrow I = \sum_{k=0}^{\infty} \frac{2\bar{a}_{2k}}{1-(2k)^2} $$

In matrix form: (Eq.2) on [[media: Egm6341.s10.mtg46.djvu | Page 46-5]], (Eq.2) on [[media: Egm6341.s10.mtg47.djvu | Page 47-3]] <br\> $$\displaystyle \bar{a}_{2k} = \frac{d_{2k}}{n} \underset{i=0}{\overset{n}{{\sum}^{\prime\prime}}} z_{2k}(\theta_i) = \mathbf b_{2k} \cdot \mathbf f$$