Exercise · ZW-COMPLEX-EXPONENTIAL-010 · ★★★ Advanced
When does \(e^{z}=e^{w}\) hold (with \(k\in\mathbb{Z}\))?
Choose one option.
Show answer
- A. When \(z-w=2\pi i k\)
Explanation
\(e^{z}=e^{w}\) is equivalent to \(e^{z-w}=1\), which happens exactly when \(z-w=2\pi i k\). The complex exponential is not injective.
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-008 Find all solutions of e^z=1 (with k∈ℤ).