Exercise · ZW-COMPLEX-LOGARITHM-002 · ★☆☆ Basic
Compute \(\operatorname{Log}(1+i)\).
Choose one option.
Show answer
- C. \(\tfrac12\ln 2+\tfrac{\pi}{4}i\)
Explanation
\(|1+i|=\sqrt2\), so \(\ln|1+i|=\tfrac12\ln2\); and \(\operatorname{Arg}(1+i)=\tfrac{\pi}{4}\).
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-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.
- ZW-COMPLEX-LOGARITHM-009 At which points is the principal logarithm Log discontinuous within its domain ℂsetminus\0\?