Fourier series functions homework

In summary, the first function, when extended as a 2pi periodic function, does not represent its Fourier series due to its lack of continuity, while the second function does represent its Fourier series due to its continuity.
  • #1
ramses07
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Homework Statement


Which of the following functions, when extended as 2pi periodic function, are equal to their Fourier series?
a. F(x) = 2x, pi<x<-pi
b. f(x)= 3 abs value (x)

Homework Equations


none



The Attempt at a Solution



After i graphed the functions into periodic functions of 2 pi, my professor told me that the first one doesn't represent its Fourier series since it doesn't converge at its end points, and the second one does. The coefficients do not need to be calculated.
Can anybody explain what my professor meant by it doesn't converge at its end points and why the b. does represent it series.
 
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  • #2
the first one, extended periodically, is not continuous while the second one is. That is, just to the left of x= pi, say x= pi- .001, the function value is 2x= 2(pi- .001) while just to the write, at x= pi+ .001, it is 2(pi+ .001- 2pi)= 2(.001- pi) because you have moved to the next period. The graph is a series of separate, parallel, line segments.

Similarly, for the secoind function, if x= pi- .001, the function value is 3|pi- .001|= 3(pi- .001) since pi> .001. If x= pi+ .001, the function value is 3|pi+ .001- 2pi|= 3|-pi+ .001|= 3|pi- .001| the same as before because of the absolute value. The graph is a series of connected line segments or "sawteeth".

At points of "jump discontinuity", the Fourier series converges to a point halfway between the two left and right limits.
 

1. What are Fourier series functions?

Fourier series functions are mathematical representations of periodic functions, such as sine and cosine waves. They can be used to decompose a complex periodic function into simpler components, making it easier to analyze and understand the behavior of the function.

2. How are Fourier series functions used in real-world applications?

Fourier series functions are used in many fields, including signal processing, image processing, and physics. They are used to analyze and manipulate signals, such as audio and radio waves, in order to extract important information or filter out unwanted components. They are also used in image compression techniques and to solve differential equations in physics.

3. What is the process for finding the Fourier series coefficients of a function?

The process for finding the Fourier series coefficients of a function involves representing the function as a sum of sine and cosine functions, known as the Fourier series. The coefficients can then be calculated using integration techniques, often using the orthogonality properties of sine and cosine functions.

4. Can Fourier series functions be used to approximate any function?

No, Fourier series functions can only approximate periodic functions. If a non-periodic function is passed through a Fourier series analysis, it will result in a periodic function with the same period as the original function. However, the approximation may not be accurate for the entire domain of the original function.

5. Are there any limitations to using Fourier series functions?

One limitation of using Fourier series functions is that they can only approximate a function within a certain period. If a function has discontinuities or sharp edges, the Fourier series may not accurately represent these features. Additionally, the accuracy of the approximation depends on the number of terms in the series, so using a larger number of terms may be necessary for more complex functions.

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