9.15 - Exercises
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Practice Questions
Test your understanding with targeted questions
What is the Fourier integral formula for f(x) = xe^{-x}, x > 0?
💡 Hint: Consider applying integration by parts.
What bounds do we use when evaluating the Fourier sine integral of a step function?
💡 Hint: Focus on where f(x) is defined.
4 more questions available
Interactive Quizzes
Quick quizzes to reinforce your learning
What is the main purpose of the Fourier cosine integral?
💡 Hint: Recall the main objective of Fourier transforms.
True or False: The Fourier sine integral is defined only for odd functions.
💡 Hint: Think about the properties of sine functions.
1 more question available
Challenge Problems
Push your limits with advanced challenges
Using the Fourier integral representation, prove the Parseval's theorem for a function defined over an infinite domain.
💡 Hint: Start by expressing both sides using the integral definitions.
Consider the function f(x) = e^{-a|x|}. Derive its full Fourier transform and discuss the significance of the results obtained in signal processing.
💡 Hint: Use properties of even functions to aid in the derivation.
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