Exercise · ZW-LIOUVILLE-THEOREM-010 · ★★★ Advanced
Let \(f\) be entire with \(f(z+1)=f(z)\) and \(f(z+i)=f(z)\) for all \(z\). What follows?
Choose one option.
Show answer
- A. \(f\) is constant
Explanation
All values of \(f\) repeat on the square \([0,1]\times[0,1]\), which is compact, so \(f\) is bounded. By Liouville it is constant. \(e^{2\pi iz}\) does not satisfy \(f(z+i)=f(z)\).
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-006 If f is entire with |f(z)|≤ M and one uses |f'(z_0)|≤ M/R for every R>0, what follows for f'(z_0)?
- 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…