Exercise · ZW-COMPLEX-EXPONENTIAL-008 · ★★☆ Intermediate
Find all solutions of \(e^{z}=1\) (with \(k\in\mathbb{Z}\)).
Choose one option.
Show answer
- C. \(z=2\pi i k\)
Explanation
We need \(e^{x}=1\), so \(x=0\), and \(\cos y=1\), \(\sin y=0\), so \(y=2\pi k\). Keeping only \(z=0\) misses the others.
Review the concept: Complex exponential
The exponential \(e^{z}\) of a complex number \(z=x+iy\) is the complex number with modulus \(e^{x}\) and argument \(y\).
\[e^{z}=e^{x}(\cos y+i\sin y)\]
More practice problems: complex exponential
- ZW-COMPLEX-EXPONENTIAL-001 Compute e^iπ.
- ZW-COMPLEX-EXPONENTIAL-002 What is the modulus of e^2+3i?
- ZW-COMPLEX-EXPONENTIAL-003 Which is an argument of e^2+3i?
- ZW-COMPLEX-EXPONENTIAL-004 Compute e^1+iπ/2.
- ZW-COMPLEX-EXPONENTIAL-005 Which is a period of e^z, that is, a number p≠0 with e^z+p=e^z for all z?
- ZW-COMPLEX-EXPONENTIAL-006 Compute e^iπ/3 e^iπ/6.
- ZW-COMPLEX-EXPONENTIAL-007 How many solutions does e^z=0 have?
- ZW-COMPLEX-EXPONENTIAL-009 Find all solutions of e^z=1+i (with k∈ℤ).