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Fourier Series

  • Thread starter Shomy
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[SOLVED] Fourier Series

1. Homework Statement
I know this may be a simple problem but im 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?


2. Homework Equations



3. The Attempt at a Solution
 

Answers and Replies

Hootenanny
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Welcome to PF Shomy,

The first thing to decide is whether the function is even, odd or neither.
 
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Thanks for the welcome!
The first part is even and the second is odd
 
Hootenanny
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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.
 
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Yeah
 
Hootenanny
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The first part is even and the second is odd
 
Hootenanny
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Odd?
 
Hootenanny
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One of the coefficients (cos) is zero
 
Hootenanny
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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?
 
Last edited:
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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?
 
Hootenanny
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Haven't we just agreed that the cosine coefficients are zero?
 
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Yeah sorry
 
Hootenanny
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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
 
Hootenanny
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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?
 
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Yeah thats what was holding up. Thanks so much!
 
Hootenanny
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Redbelly98
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> "Odd?"

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?
 
Last edited:
Hootenanny
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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:
18
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Thank You!
 
Redbelly98
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