Finding the Fourier Coefficients of a Function

In summary, the Fourier series of a function over the symmetric range [-\pi, \pi] is given by $$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^\infty a_kcos(kx)$$ where $$a_k=\frac{2}{\pi}\int^\pi_0f(x)cos(kx)dx$$
  • #1
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Homework Statement
Find the Fourier Coefficient
Relevant Equations
##f:[0,1]\rightarrow \mathbb{R}## given by
$$f(x)=x^2$$
Consider the function ##f:[0,1]\rightarrow \mathbb{R}## given by
$$f(x)=x^2$$

(1) The Fourier coefficients of ##f## are given by
$$\hat{f}(0)=\int^1_0x^2dx=\Big[\frac{x^3}{3}\Big]^1_0=\frac{1}{3}$$
$$\hat{f}(k)=\int^1_0x^2e^{-2\pi i k x}dx$$

Can this second integral be evaluated?
 
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  • #2
Why couldn't it be? I don't see what potential problem you're thinking of.
 
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  • #3
Integrate by parts twice.
 
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  • #4
@vela @pasmith thank you. Using integration by parts makes perfect sense for the integral.

I am confused about something else.

so... I know how to find the Fourier series of ##f(x)=x^2## over ##[-\pi,\pi]##, and I want to change the domain to ##[0,1]##. Is it okay to do a change of variables using ##x=\pi(2y-1)## and compute the Fourier series in ##y## space?

Not having studied the Fourier series in detail, I am asking silly questions. Could one say that ##[0,1] \subset [-\pi,\pi]## so the series over ##[-\pi,\pi]## contains the Fourier series for the other?

edit: the change of variables using ##x=\pi(2y-1)## scales the domain.
 
Last edited:
  • #5
Solution attempt:

Determining the Fourier series of a function ##f(x):[0,1]\rightarrow \mathbb{R}##

(1) Theorem: For an even function ##f(x)## over the symmetric range ##[-\pi,\pi]##, the Fourier series is given by
$$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^\infty a_k cos(kx)$$ where $$a_k=\frac{2}{\pi}\int^\pi_0f(x)cos(kx)dx$$
For ##f(x)=x^2## we compute the Fourier coefficients $$\int^\pi_0x^2dx=\Big[\frac{x^3}{3}\Big]^\pi_0=\boxed{a_0=\frac{\pi^2}{3}}$$
$$a_k=\int^\pi_0x^2cos(kx)dx=\frac{2}{\pi}\Big[\frac{2xcos(kx)}{k^2}+\Big(\frac{x^2}{k}-\frac{2}{k^3}\Big)sin(kx)\Big]^\pi_0$$
$$=\frac{2}{\pi}\Big[\frac{2xcos(kx)}{k^2}\Big]_0^\pi=\frac{4cos(kx)}{k^2}\Rightarrow \boxed{a_k=(-1)^k\frac{4}{k^2}}$$
(2) The Fourier series of ##x^2## is
$$\boxed{x^2\sim \frac{\pi^2}{3}+\sum^\infty_{n=1}(-1)^k\frac{4}{k^2}cos(kx)}$$
 
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  • #6
You found the series for ##f: [-\pi, \pi] \to \mathbb{R}##.

Did the original problem ask you to find the Fourier cosine series, in which case you would extend ##f## so you have an even function, or the Fourier series?
 
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  • #7
vela said:
You found the series for ##f: [-\pi, \pi] \to \mathbb{R}##.

Did the original problem ask you to find the Fourier cosine series, in which case you would extend ##f## so you have an even function, or the Fourier series?
Hi vela,

I'm sorry but you should to explain like I'm 5 years old what the difference is?

Thanks - newb
 
  • #8
Which of the following are you being asked to find the Fourier series of? If the question asks for the Fourier series of the function you provided in the original post, I'd interpret that as the first one. If, however, you're asked to find the Fourier cosine series, then it's understood that you extend the function so that it's an even function and find the Fourier series of that.
fourier.png

or
fourier2.png
 

What is the purpose of finding Fourier coefficients?

The purpose of finding Fourier coefficients is to represent a periodic function as a combination of simple trigonometric functions. This allows for easier analysis and manipulation of the function.

How are Fourier coefficients calculated?

Fourier coefficients are calculated using the Fourier series formula, which involves integrating the function over one period and multiplying it by trigonometric functions. This process is repeated for each coefficient.

What is the significance of the first Fourier coefficient?

The first Fourier coefficient, also known as the DC component, represents the average value of the function over one period. It is important because it determines the vertical shift of the function when it is represented as a combination of trigonometric functions.

What is the difference between Fourier series and Fourier transform?

Fourier series is used to represent a periodic function, while Fourier transform is used to represent a non-periodic function. Fourier transform also takes into account the frequency of the function, while Fourier series only considers the period.

How are Fourier coefficients used in real-world applications?

Fourier coefficients are used in various fields such as signal processing, image processing, and data compression. They are also used in the analysis of physical phenomena such as sound waves and electromagnetic waves.

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