Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 7 / b / Solution

Differentiate the prescribed imaginary part with respect to

y

:

v_{y} = e^{x} [(1 + x) cos y - y sin y] .

The Cauchy-Riemann equation

u_{x} = v_{y}

integrates to

u = e^{x} (x cos y - y sin y) + F (y) .

Its

y

u_{y} = - e^{x} [(x + 1) sin y + y cos y] + F^{'} (y)

. On the other hand,

- v_{x} = - e^{x} [(x + 1) sin y + y cos y]

. The second Cauchy-Riemann equation therefore requires

F^{'} (y) = 0

. Combining the real and imaginary parts,

e^{x} [(x cos y - y sin y) + i (y cos y + x sin y)] = (x + i y) e^{x + i y} .

Thus the most general answer on a connected domain is

f (z) = z e^{z} + C, C \in R .

A real constant preserves the given imaginary part. Conversely, the difference of any two such analytic functions has zero imaginary part, and its Cauchy-Riemann derivatives force it to be constant. This proves completeness, rather than only exhibiting one possible harmonic conjugate.

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