Half Range Fourier Series
If )
is defined on ![[0,L]](http://latex.codecogs.com/png.latex?[0,L])
then it can be extended to be either odd or even giving formulas
=\sum_{n=1}^{\infty}b_n\sin\frac{\pi nx}{L})
and
=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L})
respectively
If was not even or odd, is there another formula that can be used to express it?
I'm trying to get something of the form
=\sum_{n=0}^{\infty}c_n\sin\frac{(2n+1)\pi x}{2L})
, where 
is some constant expressed as an integral of 
, but I've no idea how to show this.
Thanks
Re: Half Range Fourier Series
The Fourier basis functions 
and 
are eigenvectors of the operator 
. Recall that eigenvectors satisfy 
. Then +b\sin(\lambda x))
. If you have (zero) Dirichlet boundary conditions, you get eigenvectors )
where 
. If you have (zero) Neumann boundary condition )
are eigenvectors. You can also have periodic boundary conditions, which yields the traditional Fourier series.
Now you ask about \over 2L} x\right))
eigenvectors. I think you can make it work with =0)
and =0)
.
Re: Half Range Fourier Series
I don't understand this, we have not yet covered eigenvectors relating to fourier series and Dirichlet boundary condition yet. Is there a simpler way to show this?
Re: Half Range Fourier Series
I think you can just break it up into odd and even parts - so  = f_{odd}(x)+f_{even}(x)=)
And by the way, =\frac{f(x)-f(-x)}{2})
and =\frac{f(x)+f(-x)}{2})
.
- Hollywood
