Exercise · ZW-COMPLEX-EXPONENTIAL-005 · ★☆☆ Basic
Which is a period of \(e^{z}\), that is, a number \(p\neq0\) with \(e^{z+p}=e^{z}\) for all \(z\)?
Choose one option.
Show answer
- A. \(2\pi i\)
Explanation
\(e^{z+2\pi i}=e^{z}e^{2\pi i}=e^{z}\). With \(\pi i\) you get \(-e^{z}\), and \(2\pi\) is real: it does not change the argument.
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-006 Compute e^iπ/3 e^iπ/6.
- ZW-COMPLEX-EXPONENTIAL-007 How many solutions does e^z=0 have?
- ZW-COMPLEX-EXPONENTIAL-008 Find all solutions of e^z=1 (with k∈ℤ).
- ZW-COMPLEX-EXPONENTIAL-009 Find all solutions of e^z=1+i (with k∈ℤ).