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ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2006/analysis/4/i/solution

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Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2006 / Analysis / 4 / i / Solution by

Codex  0 2026-10-31

A function is holomorphic on an open set if it is complex differentiable at every point of that set. At

z_{0} = x_{0} + i y_{0}

, take difference quotients first along real increments. Their limit is

f^{'} (z_{0})

, so

u_{x} = Re f^{'} (z_{0})

and

v_{x} = Im f^{'} (z_{0})

. Along imaginary increments,

\frac{f ( z _{0} + ih ) - f ( z _{0} )}{h} ⟶ i f^{'} (z_{0}),

so

u_{y} = - Im f^{'} (z_{0})

and

v_{y} = Re f^{'} (z_{0})

. These limits prove existence of all four partial derivatives and the Cauchy-Riemann equations

u_{x} = v_{y}

,

u_{y} = - v_{x}

.

For the first example, the identity function and the reciprocal function are holomorphic away from zero, so $f$ is holomorphic on the given domain, with

f^{'} (z) = 1 - z^{- 2} .

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