Cauchy--Riemann Equations Encode Complex Differentiability

Writing f equals u plus iv turns one complex derivative into two coupled real partial-derivative equations. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Statement

A holomorphic function $f:U\to\mathbf C$ has a complex derivative independent of direction. Cauchy's integral formula controls each value $f^{(n)}(a)$ from boundary data.

$$ f(z)=u(x,y)+iv(x,y),\qquad z=x+iy $$

A reliable calculation names domain and codomain. The notation $\mathsf{data}\mapsto\mathsf{claim}$ is harmless only after both $\operatorname{dom}$ and $\operatorname{cod}$ have been fixed.

$$ u_x=v_y,\qquad u_y=-v_x $$

Worked algebra

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ Df=\begin{pmatrix}u_x&u_y\\v_x&v_y\end{pmatrix}=\begin{pmatrix}a&-b\\b&a\end{pmatrix},\qquad f'(z)=a+ib $$

Conceptual compression

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f(z)=u(x,y)+iv(x,y),\qquad z=x+iy,\\[5pt] \mathsf{C}\;&:\quad u_x=v_y,\qquad u_y=-v_x. \end{aligned} $$

Caveat

Contour formulas require orientation, winding number, and hypotheses about singularities. A pole on the contour cannot be ignored.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] u_x=v_y,\qquad u_y=-v_x \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Tue 29 May 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

Cauchy--Riemann Equations Encode Complex Differentiability by Vicfred

Statement

Worked algebra

Conceptual compression

Caveat