Exercise · ZW-COMPLEX-LOGARITHM-004 · ★☆☆ Basic
What are all the logarithms of \(1\) (with \(k\in\mathbb{Z}\))?
Choose one option.
Show answer
- D. \(2\pi i k\)
Explanation
A logarithm \(w\) of \(1\) satisfies \(e^{w}=1\), which happens if \(w=2\pi i k\).
Review the concept: Complex logarithm
A logarithm of a complex number \(z\neq0\) is a complex number \(w\) whose exponential is \(z\).
\[\log z=\ln|z|+i(\arg z+2\pi k),\quad k\in\mathbb{Z}\]
More practice problems: complex logarithm
- ZW-COMPLEX-LOGARITHM-001 Compute the principal logarithm Log i.
- ZW-COMPLEX-LOGARITHM-002 Compute Log(1+i).
- ZW-COMPLEX-LOGARITHM-003 Compute Log(-1).
- ZW-COMPLEX-LOGARITHM-005 Compute Log(-e).
- ZW-COMPLEX-LOGARITHM-006 Compute Log(-2i).
- ZW-COMPLEX-LOGARITHM-007 Compute Log(e^4i).
- ZW-COMPLEX-LOGARITHM-008 With z=w=-1, compute Log(zw)-Logz-Logw.
- ZW-COMPLEX-LOGARITHM-009 At which points is the principal logarithm Log discontinuous within its domain ℂsetminus\0\?