Half Range Fourier Series

If is defined on then it can be extended to be either odd or even giving formulas

and

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

, 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 . 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 eigenvectors. I think you can make it work with and .

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

And by the way, and .

- Hollywood