Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2012 / 5 / b / i

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- Solution i

Solution

For the continuously differentiable function

f = u + i v

, the specified Wirtinger derivative is

\frac{\partial f}{\partial z ˉ} = \frac{1}{2} (u_{x} + i v_{x} + i (u_{y} + i v_{y})) = \frac{1}{2} ((u_{x} - v_{y}) + i (v_{x} + u_{y})) .

A complex number vanishes exactly when both real and imaginary parts vanish. Consequently

\frac{\partial f}{\partial z ˉ} = 0 ⟺ u_{x} = v_{y} and u_{y} = - v_{x} .

Thus the two real Cauchy-Riemann equations are one complex equation expressed by the Wirtinger derivative.

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