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Writing \( f = u + iv \), the Cauchy-Riemann equations are

  • A.\( u_{x} = v_{y} \) and \( u_{y} = v_{x} \)
  • B.\( u_{x} = -v_{y} \) and \( u_{y} = v_{x} \)
  • C.\( u_{x} = u_{y} \) and \( v_{x} = v_{y} \)
  • D.\( u_{x} = v_{y} \) and \( u_{y} = -v_{x} \)

Answer: D

Requiring the derivative to be the same whether the limit is taken along the real or the imaginary direction gives

\[ \frac{\partial u}{\partial x} \;=\; \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} \;=\; -\frac{\partial v}{\partial x}. \]

The minus sign belongs to the second pair only; putting it on the first, or omitting it altogether, describes no analytic function

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