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## Homework Statement

Let f(z) be an analytic function in the complex plane ℂ, and let [itex]\phi[/itex] be amonotonic function of a real variable.

Assume that U(x,y) = [itex]\phi[/itex](V(x,y)) where U(x,y) is the real part of f(z) and V(x,y) is the imaginary part of f(z). Prove that f is constant.

## Homework Equations

The analytic function f(z) is constant if f'(z)= 0 everywhere.

The Cauchy Riemann equation...

∂u/∂x=∂v/∂y,∂v/∂x=−∂u/∂y

## The Attempt at a Solution

I'm honestly a bit lost on where to start. I know that if f(z) is analytic then it is differentiable, so I thought that using the Cauchy Riemann equations for the partial derivatives might be helpful where U[itex]_{x}[/itex]=V[itex]_{y}[/itex] and U[itex]_{y}[/itex]=-V[itex]_{x}[/itex], but I don't know how to work with these when the function [itex]\phi[/itex] is there.

Just any information on what theory is best to look at would be helpful. Thank you.

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