Is it possible to expand a function such as $f(x) = C_0$, $C_0$ being an arbitrary positive real number, between $x = 0$ and $x = L$ in the form $$\sum_{n} C_n\sin\left(\frac{n\pi x}{L}\right)$$ with $n$ being integers?
This problem came up in a partial differential equations question where we were asked to solve $$\frac{\partial^2u}{\partial x^2} = \frac{1}{k^2}\frac{\partial u}{\partial t}$$ with boundary conditions $u(0,t) = u(L,t) = 0$ , $ u(x,0) = C_0$, I believe the solution, when the first two boundary conditions are imposed, has form $$\sum_n C_n\sin\left(\frac{n\pi x}{L}\right)e^{\frac{n^2\pi^2 k^2t}{L}}$$
and imposing the third b.c. yields
$$\sum_nC_n\sin\left(\frac{n\pi x}{L}\right) = C_0$$
and apparently $C_n$ can be derived from this, but attempting a Fourier series expansion I found the coefficients of sines to be zero and concluded that $C_0$ and $C_n$ are zero – yet apparently this is wrong.
I'm aware that the boundary conditions impose $u(0,0) = 0 = C_0$ but we were asked to disregard this as it's modeling a physical process.
Thanks in advance and sorry for any formatting issues as this is my first question here.
Apparently there's such a thing as half range Fourier series as illustrated here and I think it allows one to express the constant function as a series of sines by modeling it as only half of a periodic function.