This is an old revision of the document!
Chapter 12 Problem 9.1
Expand the following function into Legendre series. $$ f(x) = \begin{cases} -1 & -1 < x < 0 \cr 1 & 0 < x < 1 \end{cases} $$
Solution: We need to compute $$ c_n = \frac{2n+1}{2} \int_{-1}^1 f(x) P_n (x) dx $$ Then we can find that $$ f(x) \approx \sum_{n=0}^\infty c_n P_n(x). $$
To evaluate the integral, we will use the Rodrigue formula, which says $$ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n. $$
Let's use this and apply integration by part $$ \begin{aligned} \int_{-1}^1 f(x) P_n (x) dx &= - \int_{-1}^0 P_n (x) dx + \int_0^1 P_n(x) dx \cr &= \frac{1}{2^n n!} [ -\frac{d^{n-1}}{dx^{n-1}} (x^2-1)^n|^0_{-1} + \frac{d^{n-1}}{dx^{n-1}} (x^2-1)^n|^1_0 ] \cr &= \frac{1}{2^n n!} (-2) \frac{d^{n-1}}{dx^{n-1}} (x^2-1)^n (0) \end{aligned} $$
