Integrating for Fourier Series

In summary, the two given integrals can be solved using the relations sinAsinB=\frac{cos(A-B)-cos(A+B)}{2} and sinAcosB=\frac{sin(A+B)+sin(A-B)}{2}. By plotting the multiplication of two sine waves of different frequencies, the average value can be guessed graphically, which is also the result of the integration.
  • #1
tomeatworld
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Homework Statement


For positive integers m and n, calculate the two integrals:

[tex]\frac{1}{L}\int^{L}_{-L}sin(\frac{n \pi x}{L})sin(\frac{m \pi x}{L})dx[/tex] and [tex]\frac{1}{L}\int^{L}_{-L}sin(\frac{n \pi x}{L})cos(\frac{m \pi x}{L})dx[/tex]

Homework Equations


[tex]\int u v' dx = [u v] - \int u' v dx[/tex]

The Attempt at a Solution


For the first one, I can't seem to get anything other than 0 for the integral (but not in the normal way). If I work through, I end with [tex]I = \frac{n^{2}}{m^{2}}I[/tex] and that makes no sense at all. Every other part of the integral I find cancels to 0 as they all include [tex]sin(\frac{n \pi x}{L}) or sin(\frac{m \pi x}{L})[/tex] which will be 0 as n and m are integers. What am I doing wrong?

Edit: I plugged the two integrals into Mathematica and found the second one to be 0, which I calculated, but the first one is not. What do I need to change in my working?
 
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  • #2
You could use the relations

[tex] sinAsinB=\frac{cos(A-B)-cos(A+B)}{2} [/tex]

[tex] sinAcosB=\frac{sin(A+B)+sin(A-B)}{2} [/tex]

This will make your integration easy.
It is better to view graphically too. Try to plot multiplication of two sine waves of different frequencies ( a sin and cos wave of different frequencies also) and see graphically. You can guess their average value graphically. In fact that is the result of your integration also.
 

FAQ: Integrating for Fourier Series

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 allows for the decomposition of complex signals into simpler components, making it a useful tool in signal processing and analysis.

How is a Fourier series integrated?

A Fourier series is integrated by multiplying each term in the series by the corresponding sine or cosine function and integrating over one period of the function. The resulting integrals can then be solved using trigonometric identities and integration techniques.

What is the purpose of integrating for Fourier series?

The purpose of integrating for Fourier series is to find the coefficients of the sine and cosine functions that make up the series. These coefficients represent the amplitudes of the different frequency components of the original function, allowing for the analysis and manipulation of complex signals.

Are there any limitations to integrating for Fourier series?

Yes, there are limitations to integrating for Fourier series. The function being represented must be periodic, and the Fourier series may not converge for all values of x. Additionally, discontinuities and sharp corners in the function may result in inaccuracies in the Fourier series representation.

How is integrating for Fourier series used in practical applications?

Integrating for Fourier series is used in a variety of practical applications, including signal processing, image and sound compression, and data analysis. It allows for the efficient representation and manipulation of complex signals, making it a valuable tool in many scientific and engineering fields.

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