18.8.1 - Orthogonality Property
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Practice Questions
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What does orthogonality mean in the context of functions?
💡 Hint: Think about the concept of perpendicular vectors.
What are eigenfunctions?
💡 Hint: Recall their importance in solving differential equations.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the result of the inner product of two orthogonal functions?
💡 Hint: Think about what it means for vectors to be perpendicular.
True or False: Eigenfunctions can interfere when used in Fourier series.
💡 Hint: Remember, orthogonality means no overlap in contributions.
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Challenge Problems
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Prove the orthogonality of \(\sin(n\pi x/L)\) and \(\sin(m\pi x/L)\) over \([0,L]\).
💡 Hint: Utilize trigonometric identities to help simplify the integral.
Discuss the implications of the completeness property in Fourier series expansions.
💡 Hint: Consider how every piecewise continuous function can be represented in this context.
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