Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 14A / b / iv

Table of contents
- Solution iv

Solution

Use the large-

z_{0}

expansion

\frac{1}{z _{0}^{2} + R ^{2}} = \frac{1}{z _{0}} - \frac{R ^{2}}{2 z _{0}^{3}} + O (z_{0}^{- 5}) .

The first term vanishes by the compatibility condition. For the given

p

, symmetry leaves

\int p (x, y) ((x - x_{0})^{2} + (y - y_{0})^{2}), d x, d y = - 2 x_{0} (\int_{- π /2}^{π /2} x sin x d x) π = - 4 π x_{0} .

Consequently

u (x_{0}, y_{0}, z_{0}) = - \frac{x _{0}}{z _{0}^{3}} + O (z_{0}^{- 5}) .

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