I have the question:
If $u$ is a non-constant harmonic function, it must have a zero. Where is $u$ defined from the complex to the real set?
My approach has been as follows:
Since $u$ is non-constant harmonic, it must have a harmonic conjugate $v$ such that $f = u + iv$ is analytic. For a $z$ in a circle, we can use the Cauchy Integral formula to define $f(z)$ and then separate it into real and imaginary parts.
Doing this, we get the formula from the mean value property for harmonic functions. But I am very confused about the mean value property and how it ensures a zero. Since $u$ is non-constant, does it take both negative and positive values and hence by IVP, it must take zero as well? Any pointers on how to proceed?