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

Mtg 36: Thu, 12 Nov 09

page36-1

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

Example

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

End Example

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

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

Trapezoidal Rule:

 $I(f)= \int_{a}^{b}f(x)dx \$

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 $h= \frac{b-a}{n}, \ n= \$ number of panels (trapezoidal)
 $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, $E_{n}(f) \to 0 \$ as $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 $\frac{1}{r} \$ into power series in his study of attraction of spheres $\Rightarrow \$ Legendre polynomial.

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

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

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