# Half range sine series

1. Apr 28, 2016

### foo9008

1. The problem statement, all variables and given/known data
f(x) = x , 0 <x<1/2
1/2 , 1/2 < x <1
in this question , I am not convinced that a_ 0 = 0 for half range sine series , because i found that
but , thoerically , for half rang sine series , a_ 0 must be = 0 , ,am i right ? why the value of A- 0 that i got is not = 0 ? what's wrong with my working ?

2. Relevant equations

3. The attempt at a solution

2. Apr 28, 2016

### LCKurtz

Your calculation of $a_0$ is for the half range cosine series. It represents the even extension of your function. The half range sine series represents the odd extension of your function. They aren't the same thing.

3. Apr 28, 2016

### foo9008

sorry , can you explain further ? i did http://i.imgur.com/I5muGdJ.jpg
because i wanna show that for half rang sine series , a_0 = 0 , however , i get the value of a_ 0 not equal to 0 ... what's wrong with my working ?

4. Apr 28, 2016

### LCKurtz

There is nothing wrong with your calculations. They are FS of different functions. If we call the even extension of your $f(x)$ by $f_{even}(x)$, its equation on $(-1,1)$ is$$f_{even}(x) = \left \{ \begin{array}{r,l} \frac 1 2,&-1<x<-\frac 1 2\\ -x, & -\frac 1 2 < x < 0\\ x, & 0 < x < \frac 1 2 \\ \frac 1 2, &\frac 1 2 < x < 1 \end{array}\right .$$
Similarly, the odd extension $f_{odd}(x)$ is given by$$f_{odd}(x) = \left \{ \begin{array}{r,l} -\frac 1 2,&-1<x<-\frac 1 2\\ x, & -\frac 1 2 < x < \frac 1 2\\ \frac 1 2, &\frac 1 2 < x < 1 \end{array}\right .$$These are both defined on $(-1,1)$ and of period $2$. If you do the full range FS of $f_{even}$ you will find out that $b_n=0$ and you get only $a_n$ nonzero terms, and $a_0$ is what you have calculated. That's what the half range cosine series is.

If you expand $f_{odd}$ as a full range FS you will find all $a_n = 0$ and only $b_n$'s are non-zero.

The half range formulas are just shortcuts for these two series taking advantage of the even and oddness. They are two different functions. But note they both represent the same thing on $(0,1)$.

5. Apr 28, 2016

### foo9008

do you mean to get a_ 0 = 0 , we have to do the full range sine series .... so , we just need to follow the book stated that for half range sine series , a_ 0 = 0 ???
we wouldn't get a_ 0 = 0 if we do half range sine series only ?

6. Apr 28, 2016

### LCKurtz

You just need to understand that when you do a half range sine expansion of $f(x)$, you are really doing the full range expansion of $f_{odd}(x)$. But $f_{odd}(x)$ is an odd function, so its FS will have only nonzero $b_n$ and you don't have to calculate the $a_n$. The advantage of using the half range formula for $b_n$ is that the integral only involves $f(x)$ so you don't have to figure out the full formula for $f_{odd}(x)$.