Fourier Coefficients & Integrals: An Analysis

In summary, the first conversation discusses the convergence of the integral of f^2(x) on the interval -a<x<a, given that the Fourier coefficients are an=0 and bn=1/n^(1/2). The answer is that it does not converge, as the sum of the absolute values of the coefficients is infinite. The second conversation discusses finding the limit of the Fourier sine coefficients of the function f(x)=1/x on the interval -pi<x<pi as n goes to infinity. The solution involves using a change of variables to transform the integral into the form of the given hint.
  • #1
buzzmath
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Homework Statement


1. f is a function defined on the interval -a<x<a and has Fourier coefficients an=0 bn=1/n^(1/2) what can you say about the integral from -a to a of f^2(x)dx?
2. Show that as n goes to infinity the Fourier sine coefficients of the function f(x)=1/x -pi<x<pi tend to a nonzero constant. Use the fact that the integral from 0 to infinity of sin(t)/t dt=pi/2


Homework Equations





The Attempt at a Solution



1. Persevals equality says 1/a * integral from -a to a of f^2(x)dx = 2a0^2 + susm n=1 to infinity of an^2 + bn^2 so I was thinking that since to a's are zero that we could just say that f^2(x) converges to the sum of the b's but the book says the answer is f^2(x) doesn't converge since sum n=1 to infinity of (|an|+|bn|) is infinity but the a's are zero and bn = 1/n^(1/2) I thought these bn's would go to zero not infinity. What am I doing wrong?

2. f I was just going to compute the bn's from the integral directly and show that as n goes to infinity they go to a nonzero constant but I'm not sure how I would do this problem with the integral given. How does an integral from 0 to infinity come into this problem?

thanks
 
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  • #2
1. The sequence 1/n goes to 0 as n goes to infinity, but the series [itex]\sum 1/n[/itex] does not!

2. Consider doing a change of coordinate.
 
  • #3
I know that the series isn't zero but they say that the series goes to infinity which I don't understand since the sequence goes to zero. I mean how would they come up with this? is there something I'm not seeing?
What do you mean by doing a change of coordinates? how would I get integral from 0 to infinity?
 
  • #4
1. That the general term of a series goes to 0 is not a sufficient condition to assure that the series converge. The standard example used to illustrate this fact is precisely the series of 1/n, called the harmonic series: http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Divergence_of_the_harmonic_series

2. Maybe you don't see where this is going yet, but you got to try something. A logical first step seems to me to try to get the integrand in the same form as the integral in the hint. In the integralof the Fourier coefficient, you have sin(nx)/x... you'd like to get sin(t)/t. What change of variables do you suggest? And what does that leave you with?
 

Related to Fourier Coefficients & Integrals: An Analysis

1. What are Fourier coefficients and integrals?

Fourier coefficients and integrals are mathematical tools used to analyze periodic functions. They break down a periodic function into a combination of simpler trigonometric functions, allowing for easier analysis and understanding of the function's behavior.

2. How are Fourier coefficients and integrals calculated?

Fourier coefficients are calculated by taking the inner product of a periodic function with a trigonometric function. The integral of this product over one period gives the coefficient for that particular trigonometric function. Fourier integrals, on the other hand, use a complex integral to calculate the coefficients for a continuous, non-periodic function.

3. What is the significance of Fourier coefficients and integrals?

Fourier coefficients and integrals are significant because they allow us to analyze and understand complex, periodic functions in a simpler way. They have many applications in physics, engineering, and other fields where understanding periodic phenomena is important.

4. What is the difference between Fourier series and Fourier transforms?

Fourier series is used to analyze periodic functions, while Fourier transforms are used for non-periodic functions. Fourier series breaks down a periodic function into a sum of trigonometric functions, while Fourier transforms use a continuous, complex integral to break down a non-periodic function into a combination of frequency components.

5. Are Fourier coefficients and integrals only used for analyzing functions?

No, Fourier coefficients and integrals have many other applications besides function analysis. They are also used in signal processing, image processing, data compression, and solving differential equations, among other things.

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