# Holomorphisms

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Suppose that f is holomorphic in a region G(i.e. an open connected set). How can I prove that in any of the following cases

a)R(f) is constant b)I(f) is constant c)|f| is constant d) arg(f) is constant we can conclude that f is constant.

Ps. here R(f) and I(f) are the real and imaginary parts of f.

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#### Solution Preview

We only need the C-R equations.

We know, if f is analytic at a point z=(x,y), f(x,y)=u(x,y)+iv(x,y), then u,v satisfies the C-R equations: du/dx=dv/dy, du/dy=-dv/dx.

Now we come back to this problem. Since f ...

#### Solution Summary

Holomorphisms are investigated. The solution is detailed and well presented.

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