 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2013/analysis/4/a/solution

Under the assumed continuous real differentiability, the necessary and sufficient conditions are the Cauchy-Riemann equations at every point of

U

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

Necessity follows by computing the complex derivative along real and imaginary increments. For sufficiency, real differentiability gives the first-order expansion

f (z + h) - f (z) = (u_{x} + i v_{x}) Re h + (u_{y} + i v_{y}) Im h + o (∣ h ∣) .

The Cauchy-Riemann equations turn the linear part into

(u_{x} + i v_{x}) h

, so the complex derivative exists and equals

u_{x} + i v_{x}

. Thus

f

U

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