Cauchy-Riemann equations

The first-order Cauchy-Riemann operator on two real variables is ${\partial }_{\bar{z}}=\frac{1}{2}({\partial }_{x_1}+i{\partial }_{x_2})$. It is a scalar complex elliptic differential operator, since its principal symbol has nonzero modulus at every nonzero real frequency.

A prescribed inhomogeneous Wirtinger derivative equation. The distributional identity ${\partial }_{\bar{z}}(1/(\pi z))={\delta }_0$ gives its whole-plane right inverse through the Cauchy-Green operator. Boundary or decay conditions select a solution from the freedom to add a holomorphic function.

A dbar equation in a complex spectral parameter, rather than the original spatial coordinate. In spectral reconstruction, its right-hand side is transformed data. A Cauchy-Pompeiu formula in the spectral plane reconstructs the auxiliary function; an asymptotic coefficient or the original differential equation then recovers the spatial field.

For the first-order Cauchy-Riemann operator, the equation ${\varphi }_t-i{\varphi }_x=0$ admits the entire functions ${\varphi }_n(x,t)=e^{-\sqrt{n}}\cos (n(x+it))$. Every fixed-order derivative of the initial data converges uniformly to zero, since $n^r{e}^{-\sqrt{n}}\to 0$. Nevertheless ${\sup }_x|{\varphi }_n(x,t)|=e^{-\sqrt{n}}\cosh (n|t|)\to \infty$ for each $t\ne 0$. Thus even convergence of all initial derivatives does not give continuous dependence on initial data away from the initial line. Analytic existence and smooth-data stability are different properties.

For $z=x+iy$, the Wirtinger derivatives are ${\partial }_z=({\partial }_x-i{\partial }_y)/2$ and ${\partial }_{\bar{z}}=({\partial }_x+i{\partial }_y)/2$. The Cauchy-Riemann equations become ${\partial }_{\bar{z}}F=0$ for a holomorphic function $F$, and $4{\partial }_z{\partial }_{\bar{z}}=\Delta$. Thus $u_z$ is holomorphic whenever $u$ is a twice continuously differentiable harmonic function.