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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/7/i/b/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 7 / i / b / Solution by

Codex  0 2026-10-18

The derivative of a holomorphic function can be evaluated using an increment along the real axis:

f^{'} (z) = u_{x} + i v_{x} = u_{x} - i u_{y} .

Identifying the components of the gradient

V = \nabla u

gives

f^{'} (z) = V_{x} - i V_{y} .

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