Why do Fourier series require specific limits for integration?

In summary, the conversation discusses the issue of determining the appropriate limits for integrating in Fourier series. It is mentioned that the upper and lower limits are typically set as t1 and t1+T respectively, but for even functions, t1 is often chosen as -T/2. This could be due to the symmetry of the function across the y-axis. In the second case, it is explained that starting the integral from 0 allows for ending at T/2, as the integral from T/2 to T results in 0. Ultimately, the choice of t1 does not affect the result and it is best to choose the most convenient limits for integration.
  • #1
ranju
223
3

Homework Statement


The major problem I am facing while solving for Fourier series is about the limits to be taken while integrating..!
In the general equation of Fourier series the upper & lower limits are t1 & t1+T respectively..while solving for even functions we take t1 =-T/2..! Why is it so..?? does this have something to do with the symmetry across y-axis??
In the 2 attached waveforms , in the first one , limits were like -pi to pi..while in 2nd limits are 0 to T/2.>! I am not getting this..![/B]

Homework Equations


The Attempt at a Solution



The limits are t1 to t1+T whre T is the time period..but I am not getting how to decide value of t1..!
[/B]
 

Attachments

  • IMG_20140927_232752.jpg
    IMG_20140927_232752.jpg
    35.1 KB · Views: 403
  • IMG_20140927_231909.jpg
    IMG_20140927_231909.jpg
    42.4 KB · Views: 400
Physics news on Phys.org
  • #2
ranju said:
In the general equation of Fourier series the upper & lower limits are t1 & t1+T respectively..while solving for even functions we take t1 =-T/2..! Why is it so..?? does this have something to do with the symmetry across y-axis??

You have a function that is periodic with a period ##T##. From the point of view of the theory, it makes no difference what value of ##t_1## you use, it will not affect the result (try it for yourself). Therefore, it is best to take the most convenient limits for the integration. If the function is even, using ##t_1 = -T/2## could save you from the full integral, by taking twice the integral from ##t_1## to ##t_1 + T/2##.

ranju said:
In the 2 attached waveforms , in the first one , limits were like -pi to pi..while in 2nd limits are 0 to T/2.>! I am not getting this..!
The second case is somewhat similar to what I mentionned. Since the function is 0 for half the period, starting from 0 allows you to end at ##T/2##, since the integral from ##T/2## to ##T## results in 0.
 
  • Like
Likes ranju
  • #3
Ohkk.. I have got the difference..! thanks for the help...
 

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine terms. It is used to decompose a complex function into simpler components and is often used in signal processing and data analysis.

2. What is the significance of Fourier series in mathematics?

Fourier series are important because they allow us to approximate any continuous function with a series of simpler trigonometric functions. This allows us to analyze and manipulate complex functions in a more manageable way.

3. How do you calculate the coefficients of a Fourier series?

The coefficients of a Fourier series can be calculated using integration techniques, such as the Fourier series formula. This involves integrating the function over a single period and using orthogonality properties of trigonometric functions to find the coefficients.

4. What is the concept of convergence in Fourier series?

In Fourier series, convergence refers to the ability of the series to accurately approximate the original function. In other words, as we add more terms to the series, it should approach the original function. The rate of convergence depends on the smoothness of the original function.

5. Can Fourier series be used to solve differential equations?

Yes, Fourier series can be used to solve certain types of differential equations. This is because they can be used to find the general solution of a differential equation by representing the solution as a sum of trigonometric functions. This method is known as the method of undetermined coefficients.

Similar threads

  • Engineering and Comp Sci Homework Help
Replies
6
Views
3K
  • Calculus and Beyond Homework Help
Replies
3
Views
128
  • Engineering and Comp Sci Homework Help
Replies
4
Views
5K
  • Engineering and Comp Sci Homework Help
Replies
12
Views
4K
  • Engineering and Comp Sci Homework Help
Replies
26
Views
3K
  • Calculus and Beyond Homework Help
Replies
2
Views
297
  • Engineering and Comp Sci Homework Help
Replies
4
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
4
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
17
Views
4K
Back
Top