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Does anyone know how prove the following?

Suppose that $f,g$ are holomorphic functions on a non-empty open connected set $\Omega \subset \mathbb{C}$ and that $|f|^2+ |g|^2$ is constant on $\Omega$. Show that $f$ and $g$ are constant on $\Omega$.

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1 Answer 1

Note (using the Cauchy-Riemann equations) that $\Delta |f|^2 = 4 |f'|^2$. If $|f|^2 + |g|^2$ is constant, then $0 = \Delta( |f|^2 + |g|^2) = 4 (|f'|^2 + |g'|^2)$ so $f'$ and $g'$ are both $0$, and thus $f$ and $g$ are constant. Moreover, this generalizes to any number of functions.

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what is $\Delta$? –  Une Femme Douce May 7 '13 at 15:01
    
$\Delta = $ Laplacian. –  Robert Israel May 8 '13 at 6:12

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