I know that, for a central potential $U(r,\theta, \varphi)=U(r)$, the Schrödinger equation takes the well-known form in spherical coordinates. I'm searching for separable solutions in the form: \begin{equation} \phi(r, \theta,\varphi)=a(r)b(\theta)c(\varphi) \end{equation} So, if I understood correctly the theory (I'm a beginner in theoretical physics so maybe I did some conceptual mistakes) my $\phi$ is the eigenfunction corresponding to some eigenvalue E. Solving the various steps this equation appears for $c(\varphi)$: \begin{gather} \frac{d^2c}{d\varphi^2}+m_l^2=0 \end{gather} And the solution should be: \begin{equation} c(\varphi)_{m_l}=Ae^{im_l\varphi}+Be^{-im_l\varphi} \end{equation} and after some considerations on the periodicity of $\varphi$, I got that $m_l$ is integer. My problem is the following:
My teacher, and also the majority of the documents I found online, states that the solution is \begin{equation} c(\varphi)_{m_l}=Ae^{im_l\varphi} \end{equation} saying "noting that $m_l$ and $-m_l$ yield the same general solution $c(\varphi)$". What does it mean? Every solution to the harmonic oscillator differential equation yield a solution of the same form, so what is the point? It seems to me that we are "throwing away" something (even if I'm sure it's not like this, but I don't understand why). This is because I think that the equation $$\phi(r, \theta,\varphi)_{m_l}=a(r)b(\theta)c(\varphi)_{m_l}$$ would be different if I use $c(\varphi)_{m_l}=Ae^{im_l\varphi}+Be^{-im_l\varphi}$ from the one obtained by using $c(\varphi)_{m_l}=Ae^{im_l\varphi}$.
I feel like I'm missing something really stupid here but I'm in a loop now and I can't see a solution.