Homework 6, Math 471
25 September, 2008
Name:
ID:
1. Construct orthogonal polynomials of degrees 0, 1, and 2 on the interval (0,1) with
the weight function w(x) = −ln(x).
2. (a) Show by induction that for 0 ≤k≤n
dk
dxkh1−x2ni=1−x2n−kqk(x),
where qkis a polynomial of degree k.
(b) Deduce that all of the derivatives of the function (1 −x2)nof order less than n
vanish at x=±1.
(c) Define the Legendre polynomial of order jas
Pn(x) = dn
dxnh1−x2ni.
Show by repeated integration-by-parts that
hPk, Pji=Z1
−1
Pk(x)Pj(x)dx = 0,
for 0 ≤k < j .
3. (a) Show by induction that for 0 ≤k≤j
dk
dxkhxje−xi=xj−kqk(x)e−x,
where qkis a polynomial of degree k.
(b) Define the function φj, for j≥0, via
φj(x) = exdj
dxjhxje−xi.
Show that, for each j≥0, φjis a polynomial of degree j.
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