Exercise · ZW-COMPLEX-DERIVATIVE-009 · ★★★ Advanced
For \(f(z)=\bar z\) at \(z_0=2\), what happens to the difference quotient?
Choose one option.
Show answer
- A. It equals \(1\) for real \(h\) and \(-1\) for purely imaginary \(h\): there is no limit
Explanation
The quotient is \(\bar h/h\). For real \(h\) it is \(1\); for \(h=it\) it is \(-1\). Since the directional limits differ, \(\bar z\) is nowhere differentiable even though it is continuous.
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
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- 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).