# Associated Legendre polynomial (I think)

1. Oct 21, 2015

### whatisreality

1. The problem statement, all variables and given/known data
I'm not 100% sure what this type of problem is called, we weren't really told, so I'm having trouble looking it up. I'd really appreciate any resources that show solved examples, or how to find some!

Anyway. For the solution to the spherical wave equation φ(t, θ, Φ)
i) Calculate φ( 0, θ, Φ) and $\frac{\partial φ}{\partial t}$(0, θ, Φ) in terms of Anm and Bnm.

ii)Let the initial value of φ be given by φ( 0, θ, Φ) = $Y_{3}^{-2}$
and
$\frac{\partial φ}{\partial t}$(0, θ, Φ) = 16$Y_{2}^{2}$

2. Relevant equations
$P_{n}^{m}(z) = \frac{1}{2^n n!}(1-z^2)^{0.5m} \frac{d^(n+m)}{dz^(n+m)}(z^2-1)^n$ (1)

$Y_{n}^{m}(\theta, \phi) = (\frac{2n+1}{4 \pi} \frac{(n-m)!}{(n+m)!})^{0.5} e^{im \phi}P_{n}^{m}(cosθ)$ (2)

General solution:
φ = ΣΣ $A_{nm} cos(t \sqrt{n(n+1)}) + B_{nm}sin(t \sqrt{n(n+1)})$

3. The attempt at a solution

I've done part one.
φ( 0, θ, Φ) = ΣΣ $A_{nm}Y_{n}^{m}(\theta, \phi)$ from just subbing in t= 0 to the general solution.

$\frac{\partial \phi}{\partial t}(0,\theta,\phi) =$ΣΣ$B_{nm} \sqrt{n(n+1)}Y_{n}^{m}(\theta, \phi)$

But for part two, I don't know what to do with the initial values! If φ( 0, θ, Φ) = $Y_{3}^{-2}$ then does that mean m = -2 and n = 3 for the whole equation? Presumably not or $A_{nm}$ = 0.

And I can't actually evaluate $Y_{3}^{-2}$. Because subbing m and n into equation two means I have to also work out P from equation 1. So sub in z = cosθ in this case, but how can I differentiate with respect to cosθ? That doesn't make sense!

Last edited: Oct 21, 2015
2. Oct 21, 2015

### whatisreality

OK, worked out how to evaluate $Y_{3}^{-2}$, and got it equals $\sqrt{210} e^{(-2i \theta)} \frac{cos(\theta)sin(\theta)^2}{8}$, which is the equal to
∑∑ $A_{nm}Y_{n}^{m}$
Where the first sum is from n=0 to infinity and the second is from m=-n to n. So I don't know how to find $A_{nm}$ from that?