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    Fourier Series and Fourier Transform

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    1. Let f(x) be a 2pi- periodic function such that f(x) = x^2 −x for x ∈ [−pi,pi].
    Find the Fourier series for f(x).

    2. Let f(x) be a 2pi- periodic function such that f(x) = x^2 for x ∈ [−1,1]. Using
    the complex form, find the Fourier series of the function f(x).

    3. See attachment for better formula representation.
    a. Verify that the function g satisfies the condition ∫ |g(x)| ^2 dx < ∞
    b. Compute the Fourier Integral of g(x).
    c. Determine what the Fourier Integral of g(x) converges to at each real

    4. Consider the Gaussian function : (see in attachment)
    a. Sketch the graph in EXCEL of the Gaussian function when a = −0.1,
    a = 1 and a = 10 in the same frame.
    b. Compute the Fourier Transform of the Gaussian function for a = 1.

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    Solution Summary

    The solution shows, step by step, how to calculate the Fourier series and Fourier transforms of functions (including a full derivation of the transform of a Gaussian).