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## Homework Statement

Let f(z) be analytic on the set H. Let the modulus of f(z) be constant. Does f need be constant also? Explain.

## Homework Equations

Cauchy riemann equations

Hint: Prove If f and f* are both analytic on D, then f is constant.

## The Attempt at a Solution

I think f need be constant.

Let f*=conjugate operator

Let f = U+iV Then f* = U-iV

Since F is analytic we can use CR equations and we get

1) Ux = Vy

and

2) Uy = -Vx .

Applying CR to f* gives

3) Ux = -Vy

and

4) Uy = Vx

1) and 3) imply Vy = -Vy and 2) and 4) imply Vx = -Vx.

But the only function that can equal its negative is zero, and thus Vx = Vy = 0, and so V = constant.

Likewise, the same argument for Ux and Uy gives Ux = Uy = 0 and so U = a constant. And both U and V constant implies that f is constant. So we have the hint proven.

Let f = u+iv. We’re given |f| = c.

Stuck.

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