University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg33

Page 33-1

${\displaystyle \displaystyle {\begin{aligned}F:=\left\{P_{0},P_{1},P_{2},...\right\}\end{aligned}}}$	(1)

${\displaystyle \displaystyle {\begin{aligned}\left\langle f,P_{m}\right\rangle =\int _{\theta \ =-{\frac {\pi \ }{2}}}^{\frac {\pi \ }{2}}f(\theta \ )P_{m}(\sin \theta \ )\,d\theta \ \end{aligned}}}$	(2)

Where: ${\displaystyle \sin \theta \ =\mu \ \ }$ and ${\displaystyle d\theta \ \ }$ becomes ${\displaystyle d\mu \ \ }$

Similarly for ${\displaystyle \left\langle P_{n},P_{m}\right\rangle }$

Orthogonality of Legendre polynomial

${\displaystyle \displaystyle {\begin{aligned}\left\langle P_{n},P_{m}\right\rangle =\int _{-1}^{1}P_{n}(x)P_{m}(x)\,dx={\frac {2}{2n+1}}\delta \ _{mn}\end{aligned}}}$	(3)

Where ${\displaystyle \delta \ _{mn}=\ }$ kronecker delta

${\displaystyle \displaystyle {\begin{aligned}\delta \ _{mn}={\begin{cases}0&m\neq n\\1&m=n\end{cases}}\end{aligned}}}$	(4)

${\displaystyle \displaystyle {\begin{aligned}\left\langle f,P_{m}\right\rangle =\int _{-1}^{1}f(\mu \ )P_{m}(\mu \ )\,d\mu \ \end{aligned}}}$	(5)

Page 33-2

Orthogonality of ${\displaystyle \displaystyle {\begin{aligned}\left\{P_{n}\right\}=:F\end{aligned}}}$	(1)

${\displaystyle \Rightarrow \ {\underline {\Gamma \ }}(F)\ }$ is diagonal with diagonal coefficient:

${\displaystyle \displaystyle {\begin{aligned}\left\langle P_{n},P_{m}\right\rangle ={\frac {2}{2n+1}}\end{aligned}}}$	(2)

${\displaystyle \displaystyle {\begin{aligned}\Rightarrow \ A_{n}={\frac {1}{\left\langle P_{n},P_{m}\right\rangle }}\left\langle f,P_{n}\right\rangle \end{aligned}}}$	(3)

F is complete, i.e. any continuous function, f, can be expressed as an infinite series of function in F:

${\displaystyle \displaystyle {\begin{aligned}f(\mu \ )=\sum _{u=0}^{\infty }A_{n}P_{n}(\mu \ )\end{aligned}}}$	(4)

Eq(4) is an equality due to the completeness of F

p29-5: ${\displaystyle f(\theta \ )=T_{0}\cos ^{4}\theta \ =T_{0}(1-\mu \ ^{2})^{2}\sum _{u=0}^{\infty }A_{n}P_{n}(\mu \ )\ }$

Where ${\displaystyle \mu \ =\sin |theta\ \ }$

${\displaystyle \displaystyle {\begin{aligned}A_{n}={\frac {2n+1}{2}}\int _{-1}^{1}T_{0}(1-\mu \ ^{2})^{2}P_{n}(\mu \ )\,d\mu \ \end{aligned}}}$	(5)

Where n=0,1,2...n

Page 33-3

HW: ${\displaystyle f=\sum _{i}g_{i}\ }$

Show that if ${\displaystyle \left\{g_{i}\right\}\ }$ is odd, then f is odd

Show that if ${\displaystyle \left\{g_{i}\right\}\ }$ is even, then f is even

HW: Show that ${\displaystyle P_{2k}\ }$ is even for k=0,1,2... and ${\displaystyle P_{2k+1}\ }$ is odd

Eq.(5) P.33-2 ${\displaystyle A_{n}={\frac {\left\langle f,P_{n}\right\rangle }{\left\langle P_{n},P_{n}\right\rangle }}\ }$, f even

${\displaystyle \Rightarrow \ A_{n}=0\ }$ for ${\displaystyle n=2k+1\ }$, since ${\displaystyle P_{2k+1}(x)\ }$ is odd

${\displaystyle A_{1}=A_{3}=A_{5}=...=0\ }$

It turns out that ${\displaystyle A_{n}=0\ }$ for all ${\displaystyle n\geq 5\ }$ due to linear independance of ${\displaystyle F=\left\{P_{n}\right\}\ }$ and the orthogonality of ${\displaystyle F\ }$

Page 33-4

Linear independance of ${\displaystyle F\ }$

${\displaystyle P_{n}(x)\ }$ is a polynomial of order n

${\displaystyle P_{n}\in \mathrm {P} \ _{n}\ }$ set of all polynomials of degree (order) ${\displaystyle \leq \ n\ }$

${\displaystyle \displaystyle {\begin{aligned}\forall q\in \mathrm {P} \ _{n}\Rightarrow \ q(x)=\sum _{i=0}^{n}a_{i}P_{i}(x)\end{aligned}}}$	(1)

HW: ${\displaystyle q\in \mathrm {P} \ _{4}\ }$

${\displaystyle q(x)=\sum _{i=0}^{4}c_{i}x^{i}\ }$

Given ${\displaystyle c_{0}=3,c_{1}=10,c_{2}=15,c_{3}=-1,c_{4}=5\ }$

Find ${\displaystyle \left\{a_{i}\right\}\ }$ such that ${\displaystyle q(x)=\sum _{i=0}^{4}a_{i}P_{i}\ }$

Plot ${\displaystyle q=\sum _{i}c_{i}x^{i}=\sum _{i}a_{i}P_{i}\ }$

Where ${\displaystyle \sum _{i}c_{i}x^{i}=\ }$ figure 1 and ${\displaystyle \sum _{i}a_{i}P_{i}=\ }$ figure 2

Othogonality of ${\displaystyle F=\left\{P_{k}\right\}\ }$ Eq.(3) P.33-1

${\displaystyle \displaystyle {\begin{aligned}P_{m}\perp \mathrm {P} \ _{n}\ \forall m\geq \ n+1\end{aligned}}}$	(2)

References