# Lengendre Polynomials of cos(theta)

1. Mar 20, 2009

### pcalhoun

Hey guys,

I've been working on a quantum related problem in my math physics class and I've run into a snag. When dealing with Legendre Polynomials ( specifically : $$P_{lm} (x)$$ ), I can find the general expression that can be used to derive the polynomial for any sets of l and m (wolfram math has that explicit equation on their website on this page, eq 65: http://mathworld.wolfram.com/LegendrePolynomial.html" [Broken].)

However, when trying to find the general expression for the Legendre polynomial of cos: $$P_{lm}(cos(\theta))$$, I find nothing. I tried to come up with the expression on my own by substituting for $$x = cos(\theta)$$ and $$dx = -sin(\theta)d\theta$$ , but I do not know how to handle the term with the derivative which goes like: $$\frac{d^{l+m}}{dx^{l+m}}$$.

My first guess was to try and replace the differential $$dx^{l+m}$$ with $$(-sin(\theta)d\theta)^{l+m}$$, but this didn't seem to give the correct final result.

If any body knows of a website where the $$P_{lm}(cos(\theta))$$ formula is explicity given, or if someone knows how to actually derive the general form for these polynomials, let me know. I would really appreciate it. Thanks for your time.

PCalhoun

Last edited by a moderator: May 4, 2017
2. Mar 21, 2009

### tiny-tim

Hey pcalhoun!

(have a theta: θ )

d/dx = d/dθ dθ/dx = (-1/sinθ)d/dθ

d2/dx2 = (-1/sinθ)d/dθ [(-1/sinθ)d/dθ] = …

3. Mar 21, 2009

### clem

Just use (64) with cos theta=x. Then at the end go back to cos theta.