1. Ch12.3 & 12.4 (Ch11.3e & 12.4e) Orthogonality, Alternate Definitions of Legendre Polynomials
講者： 許永昌 老師

2. Contents Orthogonality Expansion of functions, Legendre series
Examples
Alternate definitions of Legendre Polynomials
Rodrigues’ Formula
Schlaefli Integral (I’ll not describe it.)

3. Orthogonality (請預讀P756~P757)
Since Pn(x) is the solution of
and L is a Hermitian Operator, we can conclude that {Pn(x)} will form an orthogonal complete set during x[-1,1].
Solve:
Gotten from the integration of g2:

4. Legendre Series (請預讀P757~P758)
Any finite and continuous function in x[-1,1] can be expanded by Legendre polynomials.
The expansion is
Usage:
an can be determined by <Pn|f> directly when f is known.
an sometimes are determined by matching boundary conditions.

5. Example 12.3.1 (請預讀P758~P759) Earth’s Gravational Field:
In fact, the Earth is pear-shaped instead of spherical.
The parameters are shown in the textbook.

6. Example 12.3.2 (請預讀P759~P761) Metal sphere in a uniform field: PDE:
B.C.:
V(r0,q,f)=0,
V(r,q,f)=E0z=-E0rcosq.
Solve:
Assuming that( RzV=V):
Substitue Eq.(2) into Eq.(1):
Fn= Anrn+Bnr-n-1,
Code: grounded_ball.m

7. Example 12.3.2 (請預讀P759~P761) Induced Surface charge:
From B.C.s, we can find
<V>=0: A0=0,
V(r,q,f) =-E0rcosq : An>1=0 & A1=-E0,
V(r0,q,f)=0 : Fn(r0)=0 and A0= An>1=0  B0= Bn>1=0
Therefore,
Induced Surface charge:
Induced dipole moment:

8. Example (請預讀P761~P762)
Electrostatic Potential of a Ring of Charge
Total Charge: q.
Radius: a .
r > a .
Solve:
Assuming that
B.C.
Comparing Eqs. (1) & (2), we get
*若直接積分還蠻麻煩的。

9. Rodrigues’ Formula (請預讀P767)
Series form:
It can be written as

10. Homework
(11.3.3e)
(11.3.7e)
(11.3.8e)
( e)
( e)
(11.4.3e)
(11.4.6e)
(11.4.7e)

11. Nouns
Rodrigues’ Formula: P767
Schlaefli Integral: P768