Half Range Fourier Series

If $\displaystyle f(x)$ is defined on $\displaystyle [0,L]$ then it can be extended to be either odd or even giving formulas

$\displaystyle f(x)=\sum_{n=1}^{\infty}b_n\sin\frac{\pi nx}{L}$ and

$\displaystyle f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L}$ respectively

If $\displaystyle f(x)$ 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

$\displaystyle f(x)=\sum_{n=0}^{\infty}c_n\sin\frac{(2n+1)\pi x}{2L}$, where $\displaystyle c_n$ is some constant expressed as an integral of $\displaystyle f$, but I've no idea how to show this.

Thanks

Re: Half Range Fourier Series

The Fourier basis functions $\displaystyle \sin$ and $\displaystyle \cos$ are eigenvectors of the operator $\displaystyle \nabla^2$. Recall that eigenvectors satisfy $\displaystyle \nabla^2 u = -\lambda^2 u$. Then $\displaystyle u = a\cos(\lambda x)+b\sin(\lambda x)$. If you have (zero) Dirichlet boundary conditions, you get eigenvectors $\displaystyle u = \sin(\lambda x)$ where $\displaystyle \lambda = {\pi n\over L}$. If you have (zero) Neumann boundary condition $\displaystyle u = \cos({\pi n\over L} x)$ are eigenvectors. You can also have periodic boundary conditions, which yields the traditional Fourier series.

Now you ask about $\displaystyle u = \sin\left({\pi (2n+1)\over 2L} x\right)$ eigenvectors. I think you can make it work with $\displaystyle u(0)=0$ and $\displaystyle {du\over dx}(L)=0$.

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 $\displaystyle f(x) = f_{odd}(x)+f_{even}(x)=$

$\displaystyle \sum_{n=1}^{\infty}b_n\sin\frac{\pi nx}{L} +\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L}=$

$\displaystyle \frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L}+b_n\sin\frac{\pi nx}{L}$

And by the way, $\displaystyle f_{odd}(x)=\frac{f(x)-f(-x)}{2}$ and $\displaystyle f_{even}(x)=\frac{f(x)+f(-x)}{2}$.

- Hollywood