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robertjford80
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Homework Statement
this comes from a problem with the Fourier series
The Attempt at a Solution
I don't get the above step.
robertjford80 said:here's another photo. I think the period is pi, but I would think cos npi would simplify to 1 not -1
A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It is used in various fields, including signal processing, physics, and engineering.
The problem with the Fourier series is that it can only accurately represent functions that are periodic and continuous. If the function is not periodic or has discontinuities, the Fourier series will have errors and may not accurately represent the original function.
The problem with the Fourier series is addressed by using a more advanced technique called the Fourier transform. This technique allows for the representation of non-periodic and discontinuous functions, making it more versatile and accurate than the Fourier series.
Gibbs phenomenon is an overshoot or ringing effect seen in the Fourier series when trying to represent a discontinuous function. It occurs due to the inability of the Fourier series to accurately represent sharp transitions in a function.
Yes, the Fourier series can only represent functions that are finite or have a finite number of discontinuities. It also has limitations when dealing with functions that have infinite slopes or are not square-integrable. These limitations are overcome by using more advanced techniques such as the Laplace transform or the Z-transform.