I'm reading the beginning of chapter 3 of Ahlfors where he starts to introduce analytic functions in regions, and I had a question on a part of Theorem 11:
Let $f=u+iv$ be an analytic function in a region $\Omega$. If the modulus of $f$ is constant, then $f$ must reduce to a constant.
His proof starts out as follows:
If $|f|=u^2+v^2$ is constant, then $ u\frac{\partial u}{\partial x} + v \frac{\partial v}{\partial x} = 0$, and $ u \frac{\partial u}{\partial y} + v \frac{\partial v}{\partial y} = -u \frac{\partial v}{\partial x} + v \frac{\partial u}{\partial x} = 0$.
My question is why does this imply that either $ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial x} = 0$ or that the determinant $u^2+v^2$ vanishes?
Best Answer
notice that you now have a solution $$\left(\begin{array}{cc} u & v\\ v & -u\end{array}\right)\left(\begin{array}{c} \frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial x}\end{array}\right)=0$$ If the modulus (which is the determinant) is not zero then the matrix is invertible, so the solution to the system above must be zero.