18.5 - Properties of Eigenfunction Expansions
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Practice Questions
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What does orthogonality mean in the context of eigenfunctions?
💡 Hint: Think about perpendicularity in geometric terms.
True or False: Completeness allows us to represent any suitable function using eigenfunctions.
💡 Hint: Consider the implications of eigenfunction sets.
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Interactive Quizzes
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What does the property of orthogonality ensure in eigenfunction expansions?
💡 Hint: Think about how distinct functions interact in an integral.
True or False: Completeness means eigenfunctions can approximate any continuous function.
💡 Hint: Consider the nature of function series.
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Challenge Problems
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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.
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.
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