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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-9/4/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 9 / 4 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

For a harmonic function

u

, put

g = u_{x} - i u_{y}

. Its Wirtinger derivative is

\frac{\partial g}{\partial z} = \frac{1}{2} (u_{xx} + u_{yy} + i (u_{x y} - u_{y x})) = 0,

so the Cauchy-Riemann equations make

g

holomorphic. In the simply connected domain it has a single-valued primitive

F (z) = \int_{z_{0}}^{z} g (ζ) d ζ .

The integral is path independent: local holomorphic primitives make its value invariant under path homotopies, and being simply connected makes any two paths with the given endpoints homotopic, as in Question 3.

Since

F^{'} = g

, the real part satisfies

(Re F)_{x} = u_{x}

and

(Re F)_{y} = Re (i g) = u_{y}

. Hence

Re F - u

is constant on the connected domain. Adding a real constant corrects it, producing a single-valued holomorphic $f$ with $Re f = u$ . The imaginary part is a harmonic conjugate and is unique up to an additive real constant. This is the global harmonic conjugate criterion by vanishing periods in the simply connected case.

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