Exercise · ZW-HOLOMORPHIC-FUNCTION-010 · ★★★ Advanced
For which pair \((u,v)\) is \(f=u+iv\) holomorphic on all of \(\mathbb{C}\)?
Choose one option.
Show answer
- D. \(u=e^x\cos y,\ v=e^x\sin y\)
Explanation
It is \(f=e^z\). In the others Cauchy-Riemann fails: \(u_x=y\neq-2y\), \(2x\neq0\), and \(1\neq-1\), respectively.
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 \(U\).
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