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

For the continuously differentiable function $f=u+iv$, the specified Wirtinger derivative isA complex number vanishes exactly when both real and imaginary parts vanish. ConsequentlyThus the two real Cauchy-Riemann equations are one complex equation expressed by the Wirtinger derivative.

The continuously differentiable function $z\bar{z}=x^2+y^2$ is a particular solution, sinceIf $f$ is any solution, set $h(z)=f(z)-z\bar{z}$. Then $h$ is continuously differentiable and its Wirtinger derivative vanishes. By the preceding part and the sufficient Cauchy-Riemann equations criterion, $h$ is a holomorphic function on all of $\mathbb{C}$, hence an entire function.

Conversely, every entire function $h$ has vanishing Wirtinger derivative, so addition of $h$ preserves the prescribed derivative. Therefore all solutions areThis is an inhomogeneous Wirtinger equation with holomorphic forcing: a particular solution plus an arbitrary element of the homogeneous solution space.