Discussion
For the complex function
\[ f(z) = f(x+iy) = x^{3}-4xy^{2}+i(4x^{2}y-2y^{3}), \]
consider the following two statements
P: f is not analytic at (0, 0).
Q: f does not satisfy the Cauchy-Riemann equations along the x-axis.
Then, which one of the following statements is true?
Answer: C
Both P and Q are true, and Q implies P.
Write \(u = x^{3}-4xy^{2}\) and \(v = 4x^{2}y-2y^{3}\). Then:
\[ \begin{aligned} u_{x} &= 3x^{2}-4y^{2}, & v_{y} &= 4x^{2}-6y^{2} \\ u_{y} &= -8xy, & v_{x} &= 8xy \end{aligned} \]
The second Cauchy-Riemann equation \(u_{y} = -v_{x}\) always holds, but \(u_{x} = v_{y}\) needs \(x^{2} = 2y^{2}\). On the x-axis (y = 0) that means x = 0, so the equations fail at every other point of the x-axis: Q is true.
Analyticity at a point needs the equations to hold throughout a neighbourhood of it. Every neighbourhood of (0, 0) contains points of the x-axis where they fail, so f is not analytic at the origin: P is true, and this argument derives P from Q, so Q implies P.
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