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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/ib/paper-2/7e/solution

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Past exam of the mathematics course of the University of Cambridge / 2001 / ib / Paper 2 / 7E / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

At a point

z_{0} = x + i y

, complex differentiability means that

f^{'} (z_{0}) = lim_{h \to 0, h \in C ∖ {0}} \frac{f ( z _{0} + h ) - f ( z _{0} )}{h}

exists and is independent of the direction of approach. Write

f = u + i v

. Restricting to real

h

gives

f^{'} = u_{x} + i v_{x}

. Restricting to

h = i t

, with real

t \to 0

, gives

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

These directional limits establish existence of the displayed partial derivatives. Equating their real and imaginary parts gives the Cauchy-Riemann equations

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

The argument applies at every point of the given open domain; it does not assume continuity of the partial derivatives beforehand.

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