Fourier Analysis Question

In summary, the conversation discusses the computation of the Fourier series of a 2pi-periodic function and the proof of a sum involving squares. The Fourier series is found for the given function, and the sum is manipulated to show that it is equivalent to a simpler sum. It is noted that this was a challenging course, but the conversation shows that the concepts were understood.
  • #1

RJLiberator

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Homework Statement


Consider a 2pi-periodic function f(x) = |x| for -pi ≤ x ≤ pi
a) Compute the Fourier series of the function f.
b) Prove that (from n=1 to n=infinity)∑ 1/(2k-1)^2 = pi^2/8.**note all "sums" from here on out will be defined from n = 1 to n=infinity

Homework Equations

The Attempt at a Solution



For part a we start with the definition of the Fourier series
f(x) = 1/2*a_0 + ∑(a_n*cos(n*x)+b_n*sin(n*x))

Since f is an even function, we know that b_n = 0.

a_n = (2/pi)*integral from 0 to pi of (x*cos(nx))dx
a_n = 2((-1)^n-1)/(pi*n^2)

a_0 = 1/pi * integral from -pi to pi of |x| dx = pi

So we have the following Fourier series for f(x) and the answer for part a:

pi/2 + ∑2((-1)^n-1)/(pi*n^2) * cos(nx)
For part b, we set x = 0, and find

|x| = pi/2 + ∑2((-1)^n-1)/(pi*n^2) * cos(nx)
=> -pi/2 = ∑2((-1)^n-1)/(pi*n^2)
=> -pi^2/4 = ∑((-1)^n-1)/n^2
Divide both sides by -2
=> pi^2/8 = ∑((-1)^n-1)/(-2*n^2)
But I'm not quite sure how to get the right side (the sum) similar to ∑ 1/(2k-1)^2
 
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  • #2
You wonder how ## {1\over 1} + 0 + {1\over 9} + 0 + {1\over 25} ... ## (n = 1, 2, 3, 4, 5) can be equal to
## {1\over 1} + {1\over 9} + {1\over 25} ... ## (k = 1, 2, 3) :smile: ?
 
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  • #3
Ah, I see.

So, in the end, I was doing things right here.

It was my inability to manipulate the problem to come to the conclusion.

Tough, tough course.
 
  • #4
Good thing there is PF :wink:
 
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What is Fourier analysis?

Fourier analysis is a mathematical technique used to break down a complex signal into simpler and more manageable components, typically represented by sine and cosine waves. It is commonly used in signal processing, image processing, and data analysis.

How does Fourier analysis work?

Fourier analysis works by representing a complex signal as a sum of simpler sine and cosine waves with different amplitudes, frequencies, and phases. This process is known as decomposition, and it allows us to understand the individual components that make up a complex signal.

What are the applications of Fourier analysis?

Fourier analysis has a wide range of applications in various fields such as engineering, physics, mathematics, and computer science. It is used in signal processing to filter out noise, in image processing to enhance image quality, and in data analysis to identify patterns and trends in data.

What are the limitations of Fourier analysis?

While Fourier analysis is a powerful tool, it does have some limitations. It assumes that the signal is periodic and stationary, meaning it repeats itself over time and does not change in shape. It also cannot handle signals that have sharp discontinuities or non-smooth changes.

What is the difference between Fourier series and Fourier transform?

Fourier series is used to decompose a periodic signal into simpler components, while Fourier transform is used to decompose a non-periodic signal into simpler components. Fourier transform also takes into account the frequency and amplitude of a signal, while Fourier series only considers the frequency.

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