- #1

lriuui0x0

- 101

- 25

- Homework Statement
- For a particular ##\lambda##, we know Legendre polynomials are the solutions to

$$

(1-x^2)\frac{d^2P}{dx^2} - 2x\frac{dP}{dx} + \lambda P = 0

$$

We can show by induction that the k-th derivative of ##P## satisfies

$$

(1-x^2)\frac{d^2P^{(k)}}{dx^2} - 2(k+1)x\frac{dP^{(k)}}{dx} + \lambda_k P^{(k)} = 0

$$

where ##\lambda_k## is fixed once ##\lambda## is fixed.

We want to show if ##P^{(k)}## is not identically zero, then ##\lambda_k \ge 0##.

- Relevant Equations
- Parseval's identity?

The problem has a hint about finding a relationship between ##\int_{-1}^1 (P^{(k+1)}(x))^2 f(x) dx## and ##\int_{-1}^1 (P^{(k)}(x))^2 g(x) dx## for suitable ##f, g##. It looks they're the weighting functions in the Sturm-Liouville theory and we may be able to make use of Parseval's identity? However I'm not sure how we exactly do this. I have recast the differential equation to the self-adjoint eigenvalue problem form:

$$

\frac{d}{dx}((1-x^2)^{k+1}\frac{dP^{(k)}}{dx}) = -\lambda_k(1-x^2)^kP^{(k)}

$$

$$

\frac{d}{dx}((1-x^2)^{k+1}\frac{dP^{(k)}}{dx}) = -\lambda_k(1-x^2)^kP^{(k)}

$$