Exercise · ZW-COMPLEX-DERIVATIVE-010 · ★★★ Advanced
For \(f(z)=|z|^2\), at which points does \(f'(z)\) exist?
Choose one option.
Show answer
- B. Only at \(z=0\)
Explanation
Since \(|z|^2=z\bar z\), the quotient at \(z_0\) is \(\bar z_0+\bar h+z_0\bar h/h\). If \(z_0\ne0\), the term \(z_0\bar h/h\) depends on the direction and there is no limit; at \(z_0=0\) the quotient is \(\bar h\to0\). Differentiable only at the origin, with \(f'(0)=0\).
Review the concept: Complex derivative
The derivative \(f'(z_0)\) of a complex function \(f\) at a point \(z_0\) is the limit of the difference quotient as \(z\) approaches \(z_0\) from any direction in the plane, if that limit exists.
\[f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}\]
More practice problems: complex derivative
- ZW-COMPLEX-DERIVATIVE-001 Compute f'(3) for f(z)=z^2.
- ZW-COMPLEX-DERIVATIVE-002 If f(z)=z^5, what is f'(1)?
- ZW-COMPLEX-DERIVATIVE-003 Compute f'(2) for f(z)=1z.
- ZW-COMPLEX-DERIVATIVE-004 What is the derivative of e^z at z=0?
- ZW-COMPLEX-DERIVATIVE-005 If f is differentiable at z_0, what follows?
- ZW-COMPLEX-DERIVATIVE-006 Compute f'(1+i) for f(z)=z^3-2z^2+5z.
- ZW-COMPLEX-DERIVATIVE-007 Compute g'(0) for g(z)=z e^z.
- ZW-COMPLEX-DERIVATIVE-008 Compute h'(1) for h(z)=(z^2)/(z+1).