Exercise · ZW-COMPLEX-EXPONENTIAL-004 · ★☆☆ Basic
Compute \(e^{1+i\pi/2}\).
Choose one option.
Show answer
- C. \(ie\)
Explanation
\(e^{1+i\pi/2}=e\,(\cos\tfrac{\pi}{2}+i\sin\tfrac{\pi}{2})=ie\).
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-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).
- ZW-COMPLEX-EXPONENTIAL-009 Find all solutions of e^z=1+i (with k∈ℤ).