In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cⁿ.

Is the absolute value function holomorphic?

As a consequence of Cauchy-Riemann equations, a real-valued holomorphic function must be constant. Therefore, the absolute value of z and the argument of z are not holomorphic.

Is constant holomorphic function?

On compact Riemann surfaces

Any holomorphic function on a compact Riemann surface is necessarily constant. , so it is constant, by the maximum modulus principle.