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3 cards from this section. Quick, good the night before a test.
What does orthogonality mean in the context of eigenfunctions?
Think about perpendicularity in geometric terms.
True or False: Completeness allows us to represent any suitable function using eigenfunctions.
Consider the implications of eigenfunction sets.
What does the property of orthogonality ensure in eigenfunction expansions?
Think about how distinct functions interact in an integral.
True or False: Completeness means eigenfunctions can approximate any continuous function.
Consider the nature of function series.
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.
Consider the concept of convergence and how functions are approached in L² space.
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?
Remember how orthogonality implies zero interaction during integration.
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