 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2014/3/a/solution

z = x + i y

, compute the complex derivative first along real increments and then along imaginary increments. Holomorphy gives

f^{'} (z) = u_{x} + i v_{x} = \frac{u _{y} + i v _{y}}{i} = v_{y} - i u_{y} .

u_{x} = v_{y}, u_{y} = - v_{x} .

The partial derivatives exist because these directional difference quotients are limits of the holomorphic derivative. This argument proves the necessary equations; it does not claim that their pointwise existence alone is sufficient for holomorphy.

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