Discussion
Which of the following is an analytic function ?
Answer: B
The analytic function is \(z^{2}\).
A function \(f = u + iv\) is analytic where it satisfies the Cauchy-Riemann equations \(u_{x} = v_{y}\), \(u_{y} = -v_{x}\). For \(z^{2} = (x^{2} - y^{2}) + i\,2xy\):
\[ \begin{aligned} u_{x} &= 2x = v_{y} \\ u_{y} &= -2y = -v_{x} \end{aligned} \]
so it is analytic everywhere. Any function that depends on \(\bar{z}\) fails. \(z + \bar{z} = 2x\) is real-valued and gives \(u_{x} = 2 \ne v_{y} = 0\). \(z\bar{z} = x^{2} + y^{2}\) satisfies the equations only at the origin. \(z/\bar{z}\) has modulus 1 everywhere and is not constant, so it cannot be analytic either.
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