Discussion
Consider two functions \(f(z) = |z|\) and \(g(z) = \bar{z}\) (conjugate of z). Using Cauchy-Riemann conditions, choose the correct answer
Answer: D
Neither f nor g is analytic.
For \(g(z) = \bar{z} = x - iy\): u = x, v = -y, so \(u_{x} = 1\) but \(v_{y} = -1\). The Cauchy-Riemann condition \(u_{x} = v_{y}\) fails everywhere, so g is nowhere differentiable.
For \(f(z) = |z|\): f is real-valued, so v = 0 and the conditions would force \(u_{x} = u_{y} = 0\), making f constant, which it is not. (At the origin |z| is not even differentiable as a real function.) Since both fail, every option claiming that one or both are analytic is wrong.
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