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

A holomorphic function on the open set $\Omega$ is complex differentiable at every point of $\Omega$: the limitexists independently of the direction of approach. Write $f(x+iy)=u(x,y)+iv(x,y)$. The Cauchy-Riemann equations areTo prove necessity, approach first with real $h$, obtaining $f^{\prime }(z)=u_x+i{v}_x$. Approach instead with $h=it$ for real $t$, obtainingEquality of real and imaginary parts yields the displayed Cauchy-Riemann equations.

A sufficient condition is that $u$ and $v$ have continuous first partial derivatives throughout $\Omega$ and satisfy the Cauchy-Riemann equations there. Indeed, real differentiability gives, for $h=a+ib$,The Cauchy-Riemann equations imply $f_y=i{f}_x$, so this is $f_{x}h+o(|h|)$. Dividing by $h$ proves complex differentiability and $f^{\prime }(z)=f_x(z)$ at every point. Merely having first partial derivatives and the equations at a point, without a suitable differentiability hypothesis, would not justify this argument.

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.