Exercise · ZW-COMPLEX-FUNCTION-009 · ★★★ Advanced
The function \(f(z)=z^2\) maps the circle \(|z|=2\) onto…
Choose one option.
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- B. \(|w|=4\)
Explanation
Since \(|z^2|=|z|^2=4\), every image point satisfies \(|w|=4\); the circle is traced twice.
Review the concept: Complex function
A rule \(f\) assigning to each \(z=x+iy\) of a set \(D\subseteq\mathbb{C}\) a unique complex number \(w=u+iv\); \(u\) and \(v\) are its real and imaginary parts.
\[f(x+iy)=u(x,y)+i\,v(x,y)\]
More practice problems: complex function
- ZW-COMPLEX-FUNCTION-001 If f(z)=z^2, compute f(1+2i).
- ZW-COMPLEX-FUNCTION-002 For f(z)=z^2 with z=x+iy, what is the real part u(x,y)?
- ZW-COMPLEX-FUNCTION-003 For f(z)=z^2 with z=x+iy, what is the imaginary part v(x,y)?
- ZW-COMPLEX-FUNCTION-004 Write f(z)=2z+bar z as u+iv with z=x+iy.
- ZW-COMPLEX-FUNCTION-005 If f(z)=1/z, compute f(1+i).
- ZW-COMPLEX-FUNCTION-006 What is the natural domain of f(z)=(1)/(z-i)?
- ZW-COMPLEX-FUNCTION-007 For f(z)=|z|^2, compute f(3+4i).
- ZW-COMPLEX-FUNCTION-008 If f(z)=z^3, compute f(1+i).