 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2012/5/b/i/solution

f = u + i v

\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.

 New to topics? Read the docs here!