Problem with the fourier series

In summary, the conversation is discussing the period of cosine and the values it can take on at different intervals. The question is asking for the smallest non-zero positive value of T that satisfies the equation cos(T+x) = cos(x), and also clarifies the values of cos(nπ) at n=1 and n=2.
  • #1
robertjford80
388
0

Homework Statement



this comes from a problem with the Fourier series

Screenshot2012-06-12at52252AMcopy.png



The Attempt at a Solution



I don't get the above step.
 
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  • #2
What is cos(0)? How about cos(π)? cos(2π)? What's the period of cosine?
 
  • #3
here's another photo. I think the period is pi, but I would think cos npi would simplify to 1 not -1
 
  • #4
I have no idea why you would think that! Surely you know that [itex]cos(\pi)= -1[/itex]?
 
  • #5
robertjford80 said:
here's another photo. I think the period is pi, but I would think cos npi would simplify to 1 not -1

The period of cos is not ∏. Think at what intervals does cos give you the same numbers it was giving for the previous interval?

In other words,

[tex]cos(T+x) = cos(x)[/tex]

For which smallest non-zero, positive value of T??Also, cos(n∏) has both the possibilities of being cos(∏) at n=1 and cos(2∏) at n=2. What are the values of each?
 

What is a Fourier series?

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.

What is the problem with the Fourier series?

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.

How is the problem with the Fourier series addressed?

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.

What is Gibbs phenomenon in relation to 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.

Are there any other limitations of the Fourier series?

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.

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