 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/19/b/solution

Let

D = (1 - x)^{2} + y^{2}

(1, 0)

, gives

u_{x} = v_{y} = \frac{2 (( 1 - x ) ^{2} - y ^{2} )}{D ^{2}}, u_{y} = - v_{x} = - \frac{4 ( 1 - x ) y}{D ^{2}} .

These are the Cauchy-Riemann equations. Equivalently, with

z = x + i y

, the analytic mapping is

u + i v = (1 + z) / (1 - z)

The supplied harmonicity result can also be checked by the multivariable chain rule. The Cauchy-Riemann equations imply

Δ u = Δ v = 0

\nabla u \cdot \nabla v = 0

, and

∣\nabla u ∣^{2} = ∣\nabla v ∣^{2}

. Hence

ΔΨ = (u_{x}^{2} + u_{y}^{2}) (Φ_{uu} + Φ_{vv}) = 0.

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