Solving a Two-Equation Fourier Series

In summary, the function is not equal to either of the other two functions. This suggests that the Fourier series coefficients are not always equal.
  • #1
Shomy
18
0
[SOLVED] Fourier Series

Homework Statement


I know this may be a simple problem but I am just beginning to understand this subject and this question has confused me.

Expand the following as a whole-range Fourier series:

f(x) = 1 -pi < x < 0
f(x) = x 0 < x < pi

I can solve FS with one eqn, but what do you do with two, do you integrate twice or something?


Homework Equations





The Attempt at a Solution

 
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  • #2
Welcome to PF Shomy,

The first thing to decide is whether the function is even, odd or neither.
 
  • #3
Thanks for the welcome!
The first part is even and the second is odd
 
  • #4
Shomy said:
The first part is even and the second is odd
I suspect that the function is written thus,

[tex]f(x) = \left\{\begin{array}{cr}1 & -\pi<x<0 \\ x & 0<x<\pi\end{array}\right.[/tex]

Which means it isn't actually two functions, but one piecewise function.
 
  • #5
Yeah
 
  • #6
Shomy said:
Yeah
So now, is the function odd or even?
 
  • #7
The first part is even and the second is odd
 
  • #8
Shomy said:
The first part is even and the second is odd
So overall the function is...?
 
  • #9
Odd?
 
  • #10
Shomy said:
Odd?
Correct! What does this tell you about the Fourier series coefficients?
 
  • #11
One of the coefficients (cos) is zero
 
  • #12
Shomy said:
One of the coefficients (cos) is zero
Correct, all the coefficients of cosine are zero. So what is the Euler formula for calculating the coefficients of sine?
 
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  • #13
Well i meant that one of the coefficients (ie the cos ) is zero because i am using sum notation.

an = 1/pi integral f(x)cosmx dx from pi to -pi. Do I integrate both functions and sum them or what?
 
  • #14
Haven't we just agreed that the cosine coefficients are zero?
 
  • #15
Yeah sorry
 
  • #16
Shomy said:
Yeah sorry
No problem :smile:. So what about the coefficients of sine?
 
  • #17
bm = 1/pi integral f(x)sinmx dx

So do you integrate the first part from pi to 0 and the second part from 0 to -pi
 
  • #18
Shomy said:
So do you integrate the first part from pi to 0 and the second part from 0 to -pi
I believe it's the other way around. Overall the coefficient is given by,

[tex]b_n=\frac{1}{\pi}\int_{0}^{2\pi}f(x)\sin(nx)dx[/tex]

However, since f(x) is defined piecewise we must write,

[tex]b_n = \frac{1}{\pi}\left[\int_{-\pi}^{0}1\cdot\sin(nx)dx + \int_{0}^{\pi}x\cdot\sin(nx)dx\right][/tex]

Is that what you meant?
 
  • #19
Yeah that's what was holding up. Thanks so much!
 
  • #20
Shomy said:
Yeah that's what was holding up. Thanks so much!
No problem :smile:
 
  • #21
> "Odd?"

Hootenanny said:
Correct! What does this tell you about the Fourier series coefficients?

If I understand the function definition, I have a problem with this answer. For example consider x = 2:

f(2) = 2
f(-2) = 1

So for x = 2,
f(x) is not equal to f(-x)
and
f(x) is not equal to -f(-x)

What does this tell us about odd vs. even vs. neither?
 
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  • #22
Good catch Redbelly! Of course the function is neither odd nor even, I really don't know what I was thinking :redface:! I'll PM Shomy now and hopefully they haven't handed the assignment in.

I've been making a few daft mistakes recently :frown:
 
Last edited:
  • #23
Thank You!
 
  • #24
Hootenanny said:
I've been making a few daft mistakes recently :frown:

It only gets worse as you get older :smile:
 

1. What is a Two-Equation Fourier Series?

A Two-Equation Fourier Series is a mathematical technique used to represent a periodic function as a combination of sine and cosine functions. It involves finding the coefficients for each term in the series, which can then be used to approximate the original function.

2. How do you solve a Two-Equation Fourier Series?

To solve a Two-Equation Fourier Series, you first need to determine the period of the function and its corresponding interval. Then, use the Fourier Series formula to calculate the coefficients for each term. Finally, the series can be expressed as a sum of these terms to approximate the original function.

3. What is the difference between a Two-Equation Fourier Series and a Fourier Transform?

A Two-Equation Fourier Series deals with periodic functions, while a Fourier Transform can be applied to both periodic and non-periodic functions. Additionally, a Fourier Transform yields a continuous spectrum of frequencies, while a Two-Equation Fourier Series only has discrete frequencies.

4. What are the applications of a Two-Equation Fourier Series?

A Two-Equation Fourier Series has many applications in science and engineering, particularly in signal processing, image analysis, and solving differential equations. It is also used in fields such as physics, chemistry, and economics to model periodic phenomena.

5. Can a Two-Equation Fourier Series represent any function?

No, a Two-Equation Fourier Series can only represent periodic functions. Non-periodic functions can be approximated using a Fourier Transform, but the resulting series will have an infinite number of terms, making it difficult to work with in practice.

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