Exercise · ZW-COMPLEX-FUNCTION-006 · ★★☆ Intermediate
What is the natural domain of \(f(z)=\dfrac{1}{z-i}\)?
Choose one option.
Show answer
- D. \(\mathbb{C}\setminus\{i\}\)
Explanation
The expression is undefined where the denominator vanishes: \(z-i=0\), that is \(z=i\).
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-007 For f(z)=|z|^2, compute f(3+4i).
- ZW-COMPLEX-FUNCTION-008 If f(z)=z^3, compute f(1+i).
- ZW-COMPLEX-FUNCTION-009 The function f(z)=z^2 maps the circle |z|=2 onto…