Fourier series representation for F(t) = 0 or sin(wt) [depending on range]

In summary, the problem involves finding the Fourier series for a function with two different intervals and a given ω. However, the ω in the function is not the same as the frequency that appears in the sine and cosine terms. The period of the function is T=4π/ω, so the fundamental frequency of the Fourier expansion is ω0=ω/2. Additionally, special treatment is needed for cases where n^2-1=0.
  • #1
Esran
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Homework Statement



Obtain the Fourier series representing the function [tex]F(t)=0[/tex] if [tex]-2\pi/w<t<0[/tex] or [tex]F(t)=sin(wt)[/tex] if [tex]0<t<2\pi/w[/tex].

Homework Equations



We have, of course, the standard equations for the coefficients of a Fourier expansion.

[tex]a_{n}=2/\tau\int^{1/2\tau}_{-1/2\tau}F(t')cos(nwt')dt'[/tex]
[tex]b_{n}=2/\tau\int^{1/2\tau}_{-1/2\tau}F(t')sin(nwt')dt'[/tex]

The Attempt at a Solution



Clearly, in the first part of the interval we're working with the integrals inside the equations for our coefficients go to zero, so we need focus only on the region [tex]0<t<2\pi/w[/tex]. The curious thing is, the problem did not mention a period, and when we assume a period of [tex]4\pi/w[/tex], each of our coefficients vanish. Should I instead assume a period of [tex]2\pi/w[/tex] and integrate over the region [tex]-\pi/w<t<\pi/w[/tex]?

Here is what I have:

For a_{n}: integral_0^((2 pi)/w)1/2 pi w cos(n w x) sin(w x) dx = (pi sin^2(pi n))/(1-n^2)

For b_{n}: integral_0^((2 pi)/w)1/2 pi w sin(w x) sin(n w x) dx = (pi sin(2 pi n))/(2 (n^2-1))

Once again, as you can see, the coefficients vanish (since n is an integer). What is going wrong here?
 
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  • #2
The ω that appears in F(t) isn't the same as the frequency that appears in the sine and cosines. As you said, the period of the function is T=4π/ω, so the fundamental frequency of the Fourier expansion is ω0=2π/T=ω/2.

Also, note that the integration results you wrote down are only valid when n2-1≠0. You need to treat cases like that separately.
 
  • #3
Oh yeah. The fundamental frequency. Thanks!
 

Related to Fourier series representation for F(t) = 0 or sin(wt) [depending on range]

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions. It is used to decompose a complex function into simpler components, making it easier to analyze and manipulate.

2. How is a Fourier series calculated?

A Fourier series is calculated by finding the coefficients of the sinusoidal functions that make up the series. These coefficients are determined by integrating the original function over one period and solving for the coefficients using the orthogonality properties of sine and cosine functions.

3. What is the range of a Fourier series?

The range of a Fourier series is determined by the period of the original function. The series will repeat itself over the period, so the range will be from 0 to the period.

4. What is the significance of using a Fourier series to represent a function?

Using a Fourier series to represent a function allows us to break down a complex function into simpler components, making it easier to analyze and manipulate. This representation is especially useful in fields such as signal processing and image analysis.

5. Can a Fourier series represent any function?

No, a Fourier series can only represent periodic functions. If a function is not periodic, it cannot be fully represented by a Fourier series. However, it is possible to use the Fourier transform to represent non-periodic functions in terms of frequency components.

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