Exercise · ZW-COMPLEX-EXPONENTIAL-002 · ★☆☆ Basic
What is the modulus of \(e^{2+3i}\)?
Choose one option.
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- B. \(e^{2}\)
Explanation
The modulus of \(e^{x+iy}\) is \(e^{x}\), which depends only on the real part; here \(x=2\).
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-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).
- ZW-COMPLEX-EXPONENTIAL-009 Find all solutions of e^z=1+i (with k∈ℤ).