Legendre Polynomials.
Basis for expansion of functions f(x) in the range -1<x<1.


The general prescription for expansion in a basis is illustrated by the Fourier series method. In the present case, our basis is the set of all Legendre polynomials, Pn(x). Then, if f(x) is an arbitrary function in -1<x<1, we write the Legendre series: wpe50.gif (1245 bytes). To find the coefficients, multiply both sides by Pn(x) and integrate over x. Due to the orthogonality and norms of the Legendre polynomials, we obtain  wpe52.gif (1367 bytes).

Exercises:

  1. Show that the coefficients for the step functionf(x) = {0 if -1<x<0 and 1 if 0<x<1}, namely wpe5A.gif (1217 bytes), are given by s0 = 1, sk=0 for k>0 even, and wpe5C.gif (2651 bytes) for k odd.
  2. Find the coefficients in the Legendre series for f(x) = |x|.
    Hints: Note that f(x) is even. Integrate by parts and use the recurrence relation. Solution: The coefficients are given by wpe53.gif (1377 bytes). To evaluate this integral, first note that the integrand has even parity if n is even and odd parity if n is odd. Therefore, the odd-n coefficients vanish and the even-n coefficients are given by wpe56.gif (1307 bytes). Second, substitute the forms of P0 and P1 into the expression for an to find a0=1/2, and a1=5/8. Lastly, wpe5E.gif (1611 bytes).

Problems:

  1. Show that wpe5F.gif (1851 bytes).
  2. Create a plot showing the step function, together with the first 6 terms of the Legendre series found in the exercise.

References


Home

Created or up-dated 08/03/99   by R.D. Poshusta
HH01580A.gif (1311 bytes)