# User:EGM6341.s11.team1.Chiu/PEA1 F09 Mtg36

Mtg 36: Thu, 12 Nov 09

Let ${\displaystyle f\in \mathbb {P} _{2n-1},\ }$ i.e. ${\displaystyle f\ }$ is a polynomial of degree ${\displaystyle \leq 2n-1\Rightarrow f^{(2n)}(x)=0\ }$

Example

 ${\displaystyle 2n-1=3\Rightarrow n=2\Rightarrow 2n=4\ }$
 ${\displaystyle f\in \mathbb {P} _{3}\Rightarrow f(x)=\sum _{j=0}^{3}c_{j}x^{j}\ }$ (1) page33-4 ${\displaystyle \Rightarrow f^{(4)}(x)=0\ }$

End Example

 ${\displaystyle f^{(2n)}(x)=0\Rightarrow E_{n}(f)=0\ }$

i.e., we can integrate exactly any polynomial of degree ${\displaystyle \leq 2n-1\ }$ using only ${\displaystyle n\ }$ integral points (almost half).

Trapezoidal Rule:

 ${\displaystyle I(f)=\int _{a}^{b}f(x)dx\ }$
 ${\displaystyle h={\frac {b-a}{n}},\ n=\ }$ number of panels (trapezoidal)
 ${\displaystyle E_{n}(f)=-{\frac {(b-a)h^{2}}{12}}f^{(2)}(\eta ),\ \eta \in \left[a,b\right]\ }$

Trapezoidal Rule can only integrate exact a straight line. Otherwise, ${\displaystyle E_{n}(f)\to 0\ }$ as ${\displaystyle n\to \infty ,\ }$ even for a simple polynomial of degree 3. (not even degree 2)

End Trapezoidal Rule

Question 1: Origin (2) of Legendre polynomial page31-3 and Legendre equation (1) page14-2.

End Question 1

Question 2: Why solving Laplace equation (heat, fluid,...) in a sphere gave rise to Legendre equation (2)?

End Question 2

Answer 1: Legendre's idea: Expand Newtonian potential ${\displaystyle {\frac {1}{r}}\ }$ into power series in his study of attraction of spheres ${\displaystyle \Rightarrow \ }$ Legendre polynomial.

Legendre found the differential equation that admits Legendre polynomial ${\displaystyle P_{n}(x)\ }$ as solutions ${\displaystyle \Rightarrow \ }$ Legendre differential equation.

Answer 2: Newtonian potentail ${\displaystyle {\frac {1}{r}}\ }$ is a solution of Laplace equation, and thus each term in power series of ${\displaystyle {\frac {1}{r}}\ }$ and thus ${\displaystyle P_{n}(x)\ }$ is also solution
${\displaystyle \Rightarrow \ }$ spherical harmonics (solution of Laplace equation in a sphere)

${\displaystyle \color {blue}\left\{Q_{n}(x)\right\}:\ }$ 2nd set of homogeneous solutions (non-polynomial) to Legendre equation.
${\displaystyle \left\{P_{n}(x)\right\}:\ }$ 1st set of homogeneous solutions (polynomial) to Legendre equation.
Legendre functions= ${\displaystyle \underbrace {\left\{P_{n}(x)\right\}} _{\color {red}polynomial}+\underbrace {\left\{Q_{n}(x)\right\}} _{\color {red}non-polynomial}\ }$
 ${\displaystyle P_{n}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Q_{n}\ }$
 ${\displaystyle P_{0}(x)=1\ \ \ \ \ \ \ \ \ \ \ Q_{0}(x)={\frac {1}{2}}\log({\frac {1+x}{1-x}})\ }$ HW4 page19-1 ${\displaystyle =\tanh ^{-1}(x)\ }$ HW