Cauchy–Riemann equations

Write $f(x+iy)=u(x,y)+iv(x,y)$ . At a point where the real partial derivatives exist, the Cauchy–Riemann equations are

u_x=v_y,\qquad u_y=-v_x.

Derivative matrix

When $f$ is real-differentiable at the point, these equations are equivalent to the real derivative matrix having the form

Df=\begin{pmatrix}a&-b\\ b&a\end{pmatrix},

so it represents multiplication by $a+ib$ .

Wirtinger notation

With

\frac{\partial}{\partial z}=\frac12\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right), \qquad \frac{\partial}{\partial\bar z}=\frac12\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right),

the equations become $\partial f/\partial\bar z=0$ . This compact notation does not remove the regularity hypotheses needed in the Cauchy–Riemann criterion.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter II, §2.