Assignment 21 - Due Thursday, June 30.

• Read section 1.3 of the Notes on Special Functions.
• Do the following problems.

• Consider the following operators, spaces, and inner products. In each case, show that the operator is self adjoint. All of these require integration by parts. See the example in section 1.3.
  1. L[f] = f′′,   V = {f in C⁽²⁾[2,4] | f(2)=0 and f(4) = 0},   < f, g > = ∫₂⁴ f(x)g(x)dx.
  2. L[f] = f′′ − f′,   V = {f in C⁽²⁾[0,∞) | f(0)=0 and f is "nice" at ∞},   < f, g > = ∫₀^∞ f(x)g(x)e^−xdx.
  3. L[f] = f′′ + (2/x)f′,   V = {f in C⁽²⁾[0,∞) | f is "nice" at 0 and f′(1)=0},   < f, g > = ∫₀¹ f(x)g(x)x²dx.

Updated 6/28/06 (fjn).