Fourier series how can it be solved?

In summary, Mark is trying to find a Fourier series for sin(n*pi). However, he is stuck on how to solve the problem. He asks for help from Hallsofivy, who is very helpful.
  • #1
KAS90
23
0

Homework Statement



find Fourier series for:
sinnπ=0
(n=0,+1,-1,+2,-2...)

I really can't understand how Fourier series works.. Like I tried solving problems..but till now, it didn't sink in my brain..
I just want 2 know the basic way of solving a problem regarding Fourier series..then it will be much easier to understand I guess..
Thanx in advance..
 
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  • #2
KAS90 said:

Homework Statement



find Fourier series for:
sinnπ=0
(n=0,+1,-1,+2,-2...)

I really can't understand how Fourier series works.. Like I tried solving problems..but till now, it didn't sink in my brain..
I just want 2 know the basic way of solving a problem regarding Fourier series..then it will be much easier to understand I guess..
Thanx in advance..

What's the function that you want to find the Fourier series for? sin (n*pi) = 0 for all integers, as you point out, but it's otherwise not very interesting. Is the function f(x) = 0?
If so, all of the coefficients in the Fourier series would be 0.

Is there more to this problem that you haven't shown us?
 
  • #3
Mark44 said:
What's the function that you want to find the Fourier series for? sin (n*pi) = 0 for all integers, as you point out, but it's otherwise not very interesting. Is the function f(x) = 0?
If so, all of the coefficients in the Fourier series would be 0.

Is there more to this problem that you haven't shown us?
hi mark
sorry 4 being late..plus giving a completely wrong question..
ok..
the I realize the problem is for example :
f(x)= sinx
0<x<pi
it's a problem I want to understand like how is it really solved?
 
  • #4
A Fourier series for a function f(x) is an infinite sum such that
[tex]f(x)= \sum_{m=0}^\infty A_n cos(nx)+ B_n sin(nx)[/tex] (or other (x/L) or whatever inside the trig functions).

Of course, the right side of that is periodic with period [itex]2\pi[/itex] so if f is not itself periodic, that can only be true on some interval (which is one reason why you might need that "/L" to alter the interval).

But if f(x)= sin(x) is not only periodic with period [itex]2\pi[/itex], it is already of that form and it is obvious that [itex]A_n= 0[/itex] for all n, while [itex]B_1= 1[/itex] and [itex]B_n= 0[/itex] for all n greater than 1.

More generally, if f(x) is integrable on the interval [0, L], then on that interval [itex]f(x)= A_n cos(2n x\pi/L)+ B_n sin(2n x\pi/L)[/itex] where
[itex]A_0= \frac{1}{L}\int_0^L f(x) dx[/itex]
[itex]A_n= \frac{1}{2L}\int_0^L f(x)cos(2n x/L)[/itex]
for n> 0 and
[itex]B_n= \frac{1}{2L}\int_0^L f(x)sin(2n x/L)[/itex].
 
  • #5
HallsofIvy said:
A Fourier series for a function f(x) is an infinite sum such that
[tex]f(x)= \sum_{m=0}^\infty A_n cos(nx)+ B_n sin(nx)[/tex] (or other (x/L) or whatever inside the trig functions).

Of course, the right side of that is periodic with period [itex]2\pi[/itex] so if f is not itself periodic, that can only be true on some interval (which is one reason why you might need that "/L" to alter the interval).

But if f(x)= sin(x) is not only periodic with period [itex]2\pi[/itex], it is already of that form and it is obvious that [itex]A_n= 0[/itex] for all n, while [itex]B_1= 1[/itex] and [itex]B_n= 0[/itex] for all n greater than 1.

More generally, if f(x) is integrable on the interval [0, L], then on that interval [itex]f(x)= A_n cos(2n x\pi/L)+ B_n sin(2n x\pi/L)[/itex] where
[itex]A_0= \frac{1}{L}\int_0^L f(x) dx[/itex]
[itex]A_n= \frac{1}{2L}\int_0^L f(x)cos(2n x/L)[/itex]
for n> 0 and
[itex]B_n= \frac{1}{2L}\int_0^L f(x)sin(2n x/L)[/itex].

THANX a lot Hallsofivy..
u were a lot of help..as usual lol..
 

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sines and cosines. It allows us to break down a complex periodic function into simpler, more manageable components.

2. How is a Fourier series solved?

To solve a Fourier series, we use a process called Fourier analysis. This involves finding the coefficients of the sines and cosines in the series, which can be done through integration and manipulation of the original function.

3. What is the use of a Fourier series?

Fourier series have many practical applications in science and engineering. They are used to analyze and synthesize signals, such as audio and radio waves, and to solve differential equations in physics and engineering.

4. Can any function be represented by a Fourier series?

No, not all functions can be represented by a Fourier series. The function must be periodic and have a finite number of discontinuities. Additionally, the function must satisfy certain mathematical criteria, such as being square integrable.

5. How accurate is a Fourier series approximation?

The accuracy of a Fourier series approximation depends on the number of terms used in the series. As the number of terms increases, the approximation becomes more accurate. However, even with an infinite number of terms, there may still be some error due to the Gibbs phenomenon.

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