 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-2/3/solution

At a point of complex differentiability, the complex derivative is independent of the direction of approach. Taking increments along the real axis gives

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

; taking imaginary increments gives

f^{'} = v_{y} - i u_{y}

u_{x} = v_{y}

and

u_{y} = - v_{x}

For the specified real part, observe that

z^{2} e^{z}

has exactly that real part. Equivalently, integrating

v_{y} = u_{x}

and checking

v_{x} = - u_{y}

gives

v = e^{x} [(x^{2} - y^{2}) sin y + 2 x y cos y] + C .

Thus

f (z) = z^{2} e^{z} + i C, C \in R .

The freedom is only a constant imaginary part: the difference of two such holomorphic functions has zero real part, so its Cauchy-Riemann equations make it constant on each connected domain.

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