Discussion
Let \(f : \mathbb{C} \to \mathbb{C}\) be defined by \(f(z) = |z|^2 - 5\bar{z} + 2\).
Which one of the following is TRUE?
Answer: B
f is differentiable at \(z = 5\).
Write \(|z|^2 = z\bar{z}\), so \(f = z\bar{z} - 5\bar{z} + 2\). A function is complex differentiable at a point exactly when its Wirtinger derivative with respect to \(\bar{z}\) vanishes there (the Cauchy-Riemann equations):
\[ \frac{\partial f}{\partial \bar{z}} = z - 5 = 0 \iff z = 5 \]
So f is differentiable at 5 and nowhere else. At \(5i\), \(-5i\) and \(-5\) the derivative \(z - 5\) is nonzero, so the Cauchy-Riemann equations fail.
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