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Fourier Series and Fourier Transform

Please show all steps.

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
number.

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 Preview

The solution is attached below in two files. the files are identical ...

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).

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