User:Egm6341.s10.team3.ynahn81/HW5-4

= (4) Find coefficients $$\displaystyle \bar{d_{2}}, \; \bar{d_{4}}, \; \bar{d_{6}}$$ from polynomials $$\displaystyle P_{2}, \; P_{4}, \; P_{6}$$ = Ref: Lecture notes [[media:Egm6341.s10.mtg27.djvu|p.27-2]]

Problem Statement
Find coefficients $$\displaystyle \bar{d_{2}}, \; \bar{d_{4}}, \; \bar{d_{6}}$$ using the following relations.


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d_{r} = \bar{d_{2r}} = \frac{P_{2r}(1)}{2^{2r}} $$ $$ 
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
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Solution
We already obtained the polynomials $$\displaystyle P_{2}(t), \; P_{4}(t)$$ in class meeting 26. (see lecture notes [[media:Egm6341.s10.mtg26.djvu|p.26-1]] and [[media:Egm6341.s10.mtg26.djvu|p.26-3]].)


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P_{2}(t) = -\frac{t^2}{2!} + \frac{1}{6} $$ $$ 
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }


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P_{4}(t) = -\frac{t^4}{24} + \frac{t^2}{12} - \frac{7}{360} $$ $$ 
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
 * }
 * }

Also, at the problem 2 in the HW5 (Ref:[[media:Egm6341.s10.mtg27.djvu|p.27-1]]), $$\displaystyle P_{6}(t) $$ have been obtained.


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P_{6}(t) = -\frac{t^6}{720} + \frac{t^4}{144} - \frac{7t^2}{720} + \frac{31}{15120} $$ $$  
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 4)
 * }
 * }

From the (Eq.1) and (Eq.2),


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$$\displaystyle d_{1} = \bar{d_{2}} = \frac{P_{2}(1)}{2^2}=\frac{1}{4}\times\frac{-3+1}{6}=-\frac{1}{12} $$  
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From the (Eq.1) and (Eq.3),


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$$\displaystyle d_{2} = \bar{d_{4}} = \frac{P_{4}(1)}{2^4}=\frac{1}{16}\times\frac{-15+30-7}{360}=\frac{1}{720} $$  
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From the (Eq.1) and (Eq.4),


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$$\displaystyle d_{3} = \bar{d_{6}} = \frac{P_{6}(1)}{2^6}=\frac{1}{64}\times\frac{-21+105-147+31}{15120}=-\frac{1}{30240} $$
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