Fourier series

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December 8th 2008, 07:19 PM
darkangel

Fourier series

Okay, I dont even know how to solve this ... plzz someone help ^^

Let f(x) be a periodic function with period P = 2pie defined in one period by

f(x) = pie if -pie < x < 0,
pie - x if 0 less than equal x less than equal pie

(a) Compute An for n greater than and equal to 0.
(b) Compute Bn for n greater than and equal to 1.
(c) Now write f(x) as a Fourier series.

I know i am asking a lot but these type of problems will be on my test -_-
Thanks
December 8th 2008, 08:35 PM
CaptainBlack

Quote:

Originally Posted by darkangel

Okay, I dont even know how to solve this ... plzz someone help ^^

Let f(x) be a periodic function with period P = 2pie defined in one period by

f(x) = pie if -pie < x < 0,
pie - x if 0 less than equal x less than equal pie

(a) Compute An for n greater than and equal to 0.
(b) Compute Bn for n greater than and equal to 1.
(c) Now write f(x) as a Fourier series.

I know i am asking a lot but these type of problems will be on my test -_-
Thanks

From the definition of a Fourier series for a function on the interval $(-\pi,\pi)$ , or periodic with period $2\pi$ :

$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx)\ dx, n=0,1,\ ...$

$b_n=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin(nx)\ dx, n=1,\ ...$

So what problems are you having with the integrals?

CB
December 9th 2008, 01:11 PM
darkangel

CaptainBlack, the problem is that our professor did not even explain us - how to solve these problems and now he is giving these on the test.

I have looked many examples, on this board. But the truth is that I dont even know how to solve these.
December 9th 2008, 08:35 PM
CaptainBlack

Quote:

Originally Posted by darkangel

CaptainBlack, the problem is that our professor did not even explain us - how to solve these problems and now he is giving these on the test.

I have looked many examples, on this board. But the truth is that I dont even know how to solve these.

The previous post gave the definition of the Fourier coeficients, to even be looking at Fourier series you must have covered integration, so you should be able to do the integrals. Further information can be found here.

But the Fourier series representation of your function is:

$f(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} [a_n\cos(nx) + b_n \sin(nx)]$

CB
December 9th 2008, 09:03 PM
darkangel

thanks captainBlack, I appreciate ur help :)
December 9th 2008, 09:34 PM
christinaaa

I have this problem as well.

The Wiki says that in order to do this problem, f(x) must be integrable from -pi to pi. This step function is not integrable? It's not a continuous function. I guess I could find the integral using geometric sums, but how would I plug that into the Fourier coefficients?
December 9th 2008, 09:48 PM
o_O

A function doesn't have to be continuous to be integralble.

$f(x) = \begin{cases} {\color{red}\pi} & {\color{red}\text{if } -\pi < x < 0} \\ {\color{blue}\pi - x} & {\color{blue}\text{if } 0 \leq x \leq \pi } \end{cases}$

f(x) is neither even nor odd so split your integral accordingly:
$\begin{aligned} a_n & =\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx)\ dx \\ & = \frac{1}{\pi} \left( \int_{-\pi}^{0} {\color{red}f(x)} \cos (nx) dx \ + \ \int_0^{\pi} {\color{blue}f(x)} \cos (nx) dx\right) \end{aligned}$
$\begin{aligned}{\color{white}a_n} & = \frac{1}{\pi} \left( \int_{-\pi}^0 {\color{red}\pi} \cos (nx) \ dx + \int_0^{\pi}{\color{blue} (\pi - x)} \cos (nx) dx\right) \end{aligned}$
December 9th 2008, 09:51 PM
christinaaa

Ah, of course! Thank you!