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