Exercise · ZW-LIOUVILLE-THEOREM-006 · ★★☆ Intermediate
If \(f\) is entire with \(|f(z)|\le M\) and one uses \(|f'(z_0)|\le M/R\) for every \(R>0\), what follows for \(f'(z_0)\)?
Choose one option.
Show answer
- C. \(f'(z_0)=0\)
Explanation
Letting \(R\to\infty\), \(M/R\to0\), so \(f'(z_0)=0\) at every point. A function with zero derivative on all of \(\mathbb{C}\) is constant; this is Liouville's proof.
Review the concept: Liouville's theorem
Liouville's theorem says that every entire function \(f\), that is, holomorphic on the whole plane, bounded by a constant \(M\) is constant.
\[|f(z)|\le M\ \ \forall z\in\mathbb{C}\ \Longrightarrow\ f\ \text{constant}\]
More practice problems: liouville's theorem
- ZW-LIOUVILLE-THEOREM-001 Which of these entire functions is bounded on all of ℂ?
- ZW-LIOUVILLE-THEOREM-002 If f is entire, |f(z)|≤5 for all z and f(0)=3, which function is f?
- ZW-LIOUVILLE-THEOREM-003 Since sin z is entire and not constant, what does Liouville give?
- ZW-LIOUVILLE-THEOREM-004 What is |cos (2i)|? Use it to see that cos z is unbounded.
- ZW-LIOUVILLE-THEOREM-005 The function (1)/(z^2+1) is bounded on the real axis. Does it contradict Liouville's theorem?
- ZW-LIOUVILLE-THEOREM-007 Which statement about entire functions is true?
- ZW-LIOUVILLE-THEOREM-008 Let f be entire with Re f(z)≤0 for all z. To which bounded function would you apply Liouville to sh…
- ZW-LIOUVILLE-THEOREM-009 Let f be entire with |f(z)|≥1 for all z. Which bounded entire function results?