How to define the Integral limits for fourier series

[Pual Black]

Homework Statement

hello
in the college we have Fourier series and i have a problem with the integral limits
i add a pdf ( 2 pages only)
my question is: how did he get the integral limits from the question
the limits are from $-\pi$ to $-\frac{\pi}{2}$ for f(x)=-2 as shown in the first page
but he changed the integral from $\frac{\pi}{2}$ to $\pi$ as shown in the second page

also the integral limits for f(x)=2 are changed but why?

 

[SteamKing]

Do you notice something special about f(x)? About how the values of f(x) are distributed about the y-axis?

 

[Pual Black]

Yes of course and it is easy to see from the figure. One of the integral limits can be taken from the figure for f(x)=+2. But the second one I didn't get it.

 

[SteamKing]

Yes of course and it is easy to see from the figure. One of the integral limits can be taken from the figure for f(x)=+2. But the second one I didn't get it.

No, it's right in front of you, but you are concentrating on one piece of the function [f(x) = 2] when you should be looking at the whole function and how it is distributed about the y-axis.

The text even tells you what this property is on page 1 when it describes f(x).

If you look carefully at the definition of a₀, you should also note another change in the integral in addition to the change in limits ...

 

[Pual Black]

The function is distributed over y-axis from -2 to +2. I see it and i know that this is an even function. $a_0$ is from zero to pi because it is half range Fourier series. The function has 3 parts to know if it is even or odd and to determine the integration limits. Right??

 

[SteamKing]

The function is distributed over y-axis from -2 to +2. I see it and i know that this is an even function. $a_0$ is from zero to pi because it is half range Fourier series. The function has 3 parts to know if it is even or odd and to determine the integration limits. Right??

More importantly, because this function is symmetric with respect to the y-axis, one only needs to integrate between the positive values of x for the key portions of the function.

The value of any integral on the negative side of the x-axis will be the same as the corresponding range on the positive x-axis; therefore, the entire integral = twice the value of the integral on the positive x-axis.

For a symmetric, or even, function f(x):

$\int^a_{-a} {f(x)}dx = 2 \int^a_0 {f(x)} dx$

Note the change in limits of integration and the multiplication by 2 of the integral on the right-hand side of the equation.

 

[Pual Black]

Woow thanks. I got it.
So if a function is even i just need to take the positive x-values for integration limits and multiply by 2.
But If the function was odd ?? How shall i take the integration limits?

Anyway i just solved the question in 2 ways. The first way was the same as in the pdf mentioned and the second way i took all 3 parts of f(x) with their integration limits and i got same answer.
Therefore taking just the positive x-values will only shorten the solution. Nothing else. Right??

 

[SteamKing]

Woow thanks. I got it.
So if a function is even i just need to take the positive x-values for integration limits and multiply by 2.
But If the function was odd ?? How shall i take the integration limits?

Knowing the definition of an odd function, you should be able to work this out for yourself.

Anyway i just solved the question in 2 ways. The first way was the same as in the pdf mentioned and the second way i took all 3 parts of f(x) with their integration limits and i got same answer.
Therefore taking just the positive x-values will only shorten the solution. Nothing else. Right??

You wanted to know why the limits of integration changed. Mission Accomplished.

Taking advantage of this characteristic of even functions reduces the amount of calculation involved in evaluating integrals over certain limit ranges, as illustrated by this example.

 

[Pual Black]

Ok thank you very much
Just another question. How can I distinguish between odd and even function.
There are many ways but is there an all around one?

 

[SteamKing]

Ok thank you very much
Just another question. How can I distinguish between odd and even function.
There are many ways but is there an all around one?

Just apply the definitions of what determines an even function and an odd function, namely:

even function:

f(-x) = f(x), for all x; f(x) is real.

odd function:

f(-x) = -f(x), for all x; f(x) is real.

http://en.wikipedia.org/wiki/Even_and_odd_functions

 

[Pual Black]

Very nice. Thank you for your help.

 

[Pual Black]

Im facing another problem. How can I distinguish between periodic functions of period $2\pi$ and half range Fourier series if he don't tell me that in the question.
It is important to know that because in 2 Pi range i have to determine all $a_0 a_n b_n$ but in half range i have to determine either $a_0 a_n$ or just $b_n$ according to the function if it is even or odd.
The $2\pi$ range can be taken from the lowest integration limit to the highest limit. But this is not always right.

 

[SteamKing]

Im facing another problem. How can I distinguish between periodic functions of period $2\pi$ and half range Fourier series if he don't tell me that in the question.
It is important to know that because in 2 Pi range i have to determine all $a_0 a_n b_n$ but in half range i have to determine either $a_0 a_n$ or just $b_n$ according to the function if it is even or odd.
The $2\pi$ range can be taken from the lowest integration limit to the highest limit. But this is not always right.

Well, you don't often get problems which are fully defined. You have to do some investigation of the function on your own and see if a period is discernible and what its value might be.