Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 7 / i

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- Solution i

Solution

For real-direction increments, complex differentiability gives

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

. For imaginary-direction increments it gives

f^{'} = (u_{y} + i v_{y}) / i = v_{y} - i u_{y}

. Equating these expressions gives the Cauchy-Riemann equations

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

A holomorphic function is smooth, so mixed partial derivatives commute. Differentiating these equations gives

u_{xx} + u_{yy} = v_{y x} - v_{x y} = 0, v_{xx} + v_{yy} = - u_{y x} + u_{x y} = 0.

Thus both real and imaginary parts are harmonic functions:

\nabla^{2} u = \nabla^{2} v = 0.

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