Assignment:
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 :
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.