User:EGM6321.F10.TEAM1.WILKS/Mtg39

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 39: Thurs, 19Nov09

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Note: Other Application: Quantum Mechanics

discrete variables <-- coding theory

Ref: [[media: http://books.google.com/books?id=4Q0ZAQAAIAAJ&q=nikiforov+functions&dq=nikiforov+functions&ei=EAsAS8yGKZbQNO-9vYkP | Nikiforov, et.al (1991) ]]

Probability (Queing theory, birth and death)

Generating functions:

-Legendre Polynomial $$ P_n \ $$: [[media: Egm6321.f09.mtg38.djvu | Eq.(5) P.38-3]]

- $$ {r \choose k} \ = $$ "r choose k" :

$$ (1+x)^r = \sum_{k=0}^\infty {r \choose k} x^k \ $$ for $$ \left | x \right \vert \le 1 \ $$

$$ A( \mu\, \rho\ ):=1-2 \mu\ \rho\ + \rho\ ^2) \ $$

From [[media: Egm6321.f09.mtg38.djvu | Eq.(6) P.38-3]] and [[media: Egm6321.f09.mtg38.djvu | Eq.(7) P.38-3]] : $$ \frac{1}{\sqrt{A( \mu\, \rho\ )}} = \alpha_0 + \alpha_1(2 \mu\ \rho\ - \rho\ ^2)+ \alpha_2(2 \mu\ \rho\ - \rho\ ^2)^2+...= \alpha_0+ (2 \mu\ \alpha_1) \rho\ + (-\alpha_1+4 \mu\ ^2 \alpha_2) \rho\ ^2 +... \ $$

Where $$ (2 \mu\ \rho\ - \rho\ ^2) = 4 \mu\ ^2 \rho\ ^2-4 \mu\ \rho\ ^3 + \rho\ ^4 \ $$

and $$ \alpha_0 = P_0( \mu\ ) \ $$

and $$ 2 \mu\ \alpha_1 = P_1( \mu\ ) \ $$

and $$ (-\alpha_1+4 \mu\ ^2 \alpha_2) = P+2( \mu\ ) \ $$

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HW Continue the power series development to find $$ P_3, P_4, P_5 \ $$ and complete result to that obatined by [[media: Egm6321.f09.mtg31.djvu | Eq.(6) P.31-3]] or [[media: Egm6321.f09.mtg31.djvu  | Eq.(7) P.31-3]] END HW

2 recurrence formulas

Plan: Find $$ \frac{d}{d \mu\ } \frac{1}{ \sqrt{A}} \ $$ and $$ \frac{d}{d \rho\ } \frac{1}{ \sqrt{A}} \ $$

1) $$\frac{d}{d \rho\ } \frac{1}{ \sqrt{A}} = -\frac{1}{2}A^{\frac{-3}{2}}\frac{dA}{d \mu\ } = \frac{ \rho\ }{A^{\frac{-3}{2}}} \ $$

Recall [[media: Egm6321.f09.mtg38.djvu | Eq.(5) P.38-3]], now  $$ \frac{d}{d \mu\ } \frac{1}{ \sqrt{A}} = \frac{ \rho\ }{A^{\frac{3}{2}}} =  \ $$

Where $$ P_0( \mu\ )=1 \Rightarrow \ P_0'=0 \ $$ and $$ P_n'( \mu\ ) = \frac{d}{d \mu\ } P_n( \mu\ ) \ $$

[[media: Egm6321.f09.mtg39.djvu | Page 39-3]]
Compare Eq(4) and Eq(5)

$$ \frac{ \rho\ ( \mu\ - \rho\ )}{A^{\frac{3}{2}}} = ( \mu\ - \rho\ ) \sum_{n=1}^\infty P_n' ( \mu\ ) \rho\ ^{n} = \rho\ \sum_{n=0}^\infty P_n ( \mu\ ) n \rho\ ^{n-1} = \rho\ \sum_{n=1}^\infty P_n { \mu\ } n \rho\ ^{n-1} $$

Where n=1 in summation due to factor u being introduced

$$ \Rightarrow \ \mu\  \sum_{n=1} P_n' \rho\  - \sum_{n=1} P_n' \rho\ ^{n-1} = \sum_{n=1} P_n \mu\ \rho\  \ $$

$$ \Rightarrow \ \mu\ P_1' \rho\ - P_1 1 \rho\ + \sum_{n=2} \left [ -P_{n-1}' + \mu\ P_n'-nP_n \right ] \rho\ ^n = 0 \ $$

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$$ \Rightarrow \ -P_{n-1}' + \mu\ P_n' - nP_n = 0 \ $$