Fourier Series Coefficients for f(x)=xcos(x) on [-∏,∏]

In summary, the user is having trouble finding the Fourier coefficients for f(x)=xcos(x) on the interval [-∏, ∏]. They have broken up the integral into two parts but keep getting a value of 0 for b1. They are asking for help and it is suggested that they may have made a mistake in their integration by not properly applying integration by parts.
  • #1
tigertan
25
0
Hey there,

First time user of this forum.

I have a question regarding an integral I've been stuck on for the past few days. I would really appreciate any eye opener into this problem!

How do I find the Fourier coefficients for f(x)=xcos(x), x [-∏,∏]

So when calculating the coefficient bn I figure that the equation I'll be working with is (1/∏)∫xcos(x)sin(kx)dx (FROM -∏ to ∏).

I keep getting down to 1/2∏∫xsin((k+1)x)dx (FROM -∏ to ∏) + 1/2∏∫xsin((k-1)x)dx (FROM -∏ to ∏) . I some how always get a 0 for b1. What am I doing incorrectly??
 
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  • #2
Welcome to PF, tigertan! :smile:

I'm not getting 0 for b1.
So what did you do?
 
  • #3
Am I right in breaking it up to 1/2∏∫xsin((k+1)x)dx (FROM -∏ to ∏) + 1/2∏∫xsin((k-1)x)dx (FROM -∏ to ∏)?

I then end up with 1/2∏[-(k+1)cos((k+1)x) + sin((k+1)x)](FROM -∏ to ∏) + 1/2∏[-(k-1)cos((k-1)x) + sin((k-1)x)](FROM -∏ to ∏)

This equals to 0 when I put b1
 
  • #4
You lost a factor x in your integration.
It seems you did not properly apply integration by parts.

You should have an expression containing: x cos(k+1)x.
When you substitute the bounds -pi and +pi, these add up.
 

1. What is the Fourier Series Integral?

The Fourier Series Integral is a mathematical tool that allows us to represent a periodic function as a sum of sine and cosine functions.

2. What is the purpose of using Fourier Series Integral?

The purpose of using Fourier Series Integral is to break down a complex periodic function into simpler components, making it easier to analyze and manipulate mathematically.

3. How is the Fourier Series Integral calculated?

The Fourier Series Integral is calculated by finding the coefficients of the sine and cosine functions that best fit the given periodic function. This involves using complex mathematical techniques such as integration and orthogonality.

4. What are some applications of Fourier Series Integral?

Fourier Series Integral has various applications in fields such as signal processing, image and sound compression, and solving differential equations. It is also used in physics and engineering to model and analyze periodic phenomena.

5. Can any periodic function be represented using Fourier Series Integral?

Yes, any periodic function with a finite number of discontinuities and a finite number of maxima and minima can be represented using Fourier Series Integral. However, the convergence of the series may vary depending on the function.

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