18.5. Properties of Eigenfunction Expansions

Try these first

1. 1.

   What does orthogonality mean in the context of eigenfunctions?

   Hint

   Think about perpendicularity in geometric terms.

2. 2.

   True or False: Completeness allows us to represent any suitable function using eigenfunctions.

   Hint

   Consider the implications of eigenfunction sets.

3. 3.

   What does the property of orthogonality ensure in eigenfunction expansions?

   • Increased complexity
   • Simplified coefficient computation
   • No effect on calculations

   Hint

   Think about how distinct functions interact in an integral.

4. 4.

   True or False: Completeness means eigenfunctions can approximate any continuous function.

   • True
   • False

   Hint

   Consider the nature of function series.

5. 5.

   Prove that the set of eigenfunctions {sin(nπx/L)} for n=1,2,3,... on the interval [0,L] is complete for the space of square-integrable functions.

   Hint

   Consider the concept of convergence and how functions are approached in L² space.

6. 6.

   If the eigenfunctions φ₁(x) and φ₂(x) are orthogonal, and you know their eigenvalues are λ₁ and λ₂ respectively, under what circumstances would their coefficients not affect each other?

   Hint

   Remember how orthogonality implies zero interaction during integration.