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.

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).

Fourier series analysis based on examining shape of given waveforms. ... A Fourier series analysis is considered based on waveform shapes and symmetries only. 4. Q1. ...

... a series of continuous function periodic sine(f) in the Fourier series to simulate ... The second part explains the Fourier transform of a continuous signal sinc(t ...

... Hi I have developed the Fourier Series analysis section further so please see revised Word Doc. Complex maths so needs checking in detail for any mistakes. ...

... The first term has the Inverse Laplace Transform (IFT) of the step ... Time (s) - Fourier Series representation a worked example Any periodic waveform, period T ...

... By definition of Fourier transform. ... R = 2 Ω, C = 1/ 2 F, and L = 4 H . The initial 4. Consider a series RLC circuit with voltage across the capacitor is 2 V ...

... The protocol is a series of tests. ...Fourier transform Infrared spectroscopy; Microspectrophotmetry; Scanning electron microscopy; Pyrolysis coupled with gas ...

... d,q) time series models o Further sources of information o Exercises Time series analysis II: frequency-domain o Fourier and Fast Fourier Transforms FT and ...