User:EGM6341.s11.TEAM1.WILKS/Mtg39

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_11_15_13_58.djvu | Mtg 39:]] Thu, 11 Nov 10

= Family of 2nd homogeneous solutions of Legendre equation =

[[media: 2010_11_11_15_13_58.djvu | Page 39-1]]
$$ \left \{ Q_0, Q_1,... \right \} \ $$ non-polynomial

Recall the 1st homogeneous solutions: $$ \left \{ P_0, P_1,... \right \} \ $$ polynomials

$$ \left \{ P_n \right \} \oplus \left \{ Q_n \right \} = \ $$ Legendre functions $$ P_0(x)=1 \ $$ HW7.5

$$ Q_0(x)=\frac{1}{2} \log \left( \frac{1+x}{1-x} \right) = \tanh ^{-1}x \ $$ F09 HW4

$$ Q_1(x) = \frac{x}{2} \log \left( \frac{1+x}{1-x} \right) - 1 = x \tanh ^{-1} x - 1 \ $$ F09 HW4

END HW7.5

See (K.p.33) for $$ Q_2, Q_3,... \ $$ for $$ Q_2 \ $$ see [[media: 2010_11_02_14_58_42.djvu | (p.32-3)]] and HW6.1

In general,

[[media: 2010_11_11_15_13_58.djvu | Page 39-2]]
where
 * {| style="width:100%" border="0"

$$  \displaystyle \left[ \cdot \right] = \text{ Int}\left( \cdot \right) $$     (1b) means the "integer part of"; the notation $$\displaystyle \text{ Int}\left( \cdot \right)$$ is less confusing than the notation $$\displaystyle [\cdot]$$, which could be interpreted as simple square brackets. To avoid such confusion, the floor brackets could be used to indicate the integer part:
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \lfloor \cdot \rfloor := \text{ Int}\left( \cdot \right) $$     (1c) See Floor and ceiling functions (wikipedia). Still, it's hard to beat the clarity of the notation $$\displaystyle \text{ Int}\left( \cdot \right)$$.
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

$$ Q_0(x) = \tanh ^{-1}x \ $$ is odd. HW*7.6:

Use [[media: 2010_11_11_15_13_58.djvu | Eq.(1)p.39-1]] to show $$ Q_n \ $$ even or odd depending on "n"

End HW*7.6

= Orthogonality of Legendre functions, proof =

Legendre functions: $$ L_n(x)=P_n(x) \ $$ OR $$ Q_n(x) \ $$

[[media: 2010_11_11_15_13_58.djvu | Page 39-3]]
Proof:

[http://books.google.com/books?id=9Cg3HWCnCjAC&printsec=frontcover&dq=differential+equations+billingham&ei=pGR4SpPVLojSMpb07Qw#v=onepage&q=&f=false cf. K.2003 p.41-42], where the Rodrigues's formula is used. Here, we show [[media: 2010_11_11_15_13_58.djvu | Eq.(3)p.39-2]] using Legendre [[media: 2010_11_05_07_46_02.djvu | Eq.(1)and Eq.(3) p.35-3]] $$ \equiv \ \ $$ [[media: 2010_09_02_13_55_50.djvu | Eq.(1) p.5-4]] $$ \equiv \ \ $$ [[media: 2010_10_14_14_02_27.djvu | Eq.(1) p.24-1]] where $$ \left [ (1-x^2)y' \right ]' = (1-x^2)y''-2xy' \ $$ $$ \Rightarrow \ \left [ (1-x^2)L_n' \right ]'+ n(n+1)L_n=0 \ $$ Multiply by $$ L_m \ $$ and integrate over $$ \left [ -1,+1 \right ] \ $$ $$ \int_{-1}^{1} L_m \left [ (1-x^2)L_n' \right ]' \, dx +n(n+1)\int_{-1}^{1} L_m L_n\, dx  \ $$ Integrate by parts $$ \int_{-1}^{1} L_m \left [ (1-x^2)L_n' \right ]' \, dx = \alpha\ \ $$ to obtain where $$ \int_{-1}^{1} L_m L_n, dx = \left \langle L_m,L_n \right \rangle  \ $$ from F09

[[media: 2010_11_11_15_13_58.djvu | Page 39-4]]
Interchange n and m: Now subtract Eq.(3) from Eq.(2) to get $$ \left [ n(n+1)-m(m+1) \right ] \left \langle L_n,L_m \right \rangle =0 \ $$ Where $$ \left [ n(n+1)-m(m+1) \right ] \ne 0 \ $$ since $$ m \ne n  \ $$ $$ \Rightarrow \ \left \langle L_n,L_m \right \rangle =0 \ $$ because $$ m \ne n \ $$ END Proof

= Historical origin of the Legendre functions =

NOTE:

Historical development of $$ \left \{ P_n \right \} \ $$, where $$ P_n \ $$ is a Legendre polynomial. Q1: Origin of $$ \left \{ P_n \right \} \ $$ and Legendre equation [[media: 2010_09_02_13_55_50.djvu | Eq.(1) p.5-4]] Q2: Why solving Laplace equation (heat, fluids, electromagnetics, ...) in a sphere gave rise to Legendre Equation?

[[media: 2010_11_11_15_13_58.djvu | Page 39-5]]
A1: Legendre idea: Study attraction of spheres $$ \Rightarrow \ \ $$ expand Newtonian potential $$ \frac{1}{r} \ $$ in power series $$ \Rightarrow \ \ $$ Found Legendre polynomial $$ \left \{ P_n \right \} \ $$.

Studied properties of $$ \left \{ P_n \right \} \Rightarrow \ \ $$ found difference (recurrence) equation, then differential (Legendre) equation governing $$ \left \{ P_n \right \} \ $$. A2: Newtonian potential $$ \frac{1}{r} \ $$ is a solution of Laplace equation $$ \Delta\ \frac{1}{r}=0 \ $$, thus each term of power series of $$ \frac{1}{r} \ $$, and thus $$ P_n(x) \ $$ is also a solution. (superposition and induction) $$ \Rightarrow \ \ $$ spherical harmonics (solution of Laplace equation in a sphere) END NOTE