My book gives an example of solving Laplace's equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ on a rectangle of length $L$ and height $H$ with boundary conditions $u(x,0) = 0, u(x,H) = f(x), u(0,y) = 0, u(L,y) = 0$ . It explains the procedure as follows :
To solve this equation, we use separation of variables. We get $\frac{X''}{X} = -\frac{Y''}{Y} = \lambda = \pm k^2$. The only case that gives a non trivial solution is when $\lambda < 0$. It follows that $X(x) = Acos(kx) + Bsin(kx)$. One of the homogeneous boundary conditions gives that $sin(kL) = 0$. This equation is verified for values of $kL = n\pi$ with $n = 1,2,...,\infty$. We do not include $n =0$ because it correspond to the case $\lambda = 0$ which gives a trivial solution...
My question : Why the author did not consider the negatives values of $n$?