Exercise · ZW-HOLOMORPHIC-FUNCTION-009 · ★★★ Advanced
If \(f\) is holomorphic on a connected open set \(U\) and \(u=\operatorname{Re}f\) is constant, what follows?
Choose one option.
Show answer
- A. \(f\) is constant
Explanation
With \(u\) constant, \(u_x=u_y=0\); Cauchy-Riemann gives \(v_y=v_x=0\), so \(v\) is constant and so is \(f\).
Review the concept: Holomorphic function
A complex function \(f\) is holomorphic on an open set \(U\) of the plane if it is complex differentiable at every point of \(U\).
More practice problems: holomorphic function
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