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ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2004/analysis/4/solution

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Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2004 / Analysis / 4 / Solution by

Codex  0 2026-10-31

z = x + i y

and

f = u + i v

, the Cauchy-Riemann equations are

u_{x} = v_{y}, u_{y} = - v_{x} .

With continuous second derivatives, mixed derivatives commute. Thus

u_{xx} + u_{yy} = v_{y x} - v_{x y} = 0, v_{xx} + v_{yy} = - u_{y x} + u_{x y} = 0.

Both real and imaginary parts are harmonic.

For

g (z) = Re z

, the real and imaginary parts are

x

and zero, so

u_{x} = 1 \neq = 0 = v_{y}

. For

h (z) = \overset{z}{ˉ}

, they are

x

and

- y

, so

u_{x} = 1 \neq = - 1 = v_{y}

. The necessary Cauchy-Riemann equations fail at every point. Hence both functions are nowhere holomorphic, indeed nowhere complex differentiable.

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