Exercise · ZW-COMPLEX-LOGARITHM-009 · ★★★ Advanced
At which points is the principal logarithm \(\operatorname{Log}\) discontinuous within its domain \(\mathbb{C}\setminus\{0\}\)?
Choose one option.
Show answer
- B. On the negative reals
Explanation
\(\operatorname{Arg}\) jumps from near \(\pi\) to near \(-\pi\) when crossing the negative real axis, so \(\operatorname{Log}\) jumps there by about \(2\pi i\).
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-004 What are all the logarithms of 1 (with k∈ℤ)?
- 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.