A complex function $f$ with real and imaginary parts $u$ and $v$ respectively is holomorphic in some domain $\Omega$ iff $u$ and $v$ satisfy the Cauchy-Riemann equations in $\Omega$:
$$\frac {\partial u}{\partial x} = \frac {\partial v}{\partial y} \\ \frac {\partial v}{\partial x} = -\frac {\partial u}{\partial y}.$$ I am looking for different ways of recalling or producing this result. One way is to begin with the requirement $\bar \partial f= 0,$ with $\bar \partial = \partial_x + i \partial_y.$ The real and imaginary parts of $\partial_x(u+iv) + i\partial_y(u+iv) = 0$ are then the Cauchy-Riemann equations.
Another well-known heuristic is to compare the Jacobian $\begin{pmatrix} u_x & u_y\\ v_x & v_y\end{pmatrix}$ with the matrix representation of a complex number. How else do you produce these equations when needed?
Best Answer
If $\frac{\partial f}{\partial z}$ is well-defined for complex $z$, then for real $x$ and $y,$ $$ \frac{\partial f}{\partial x} = \frac{\partial f}{\partial(iy)} = -i\frac{\partial f}{\partial y} $$ That is, $$ \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} =-i\left(\frac{\partial u}{\partial y} + i\frac{\partial v}{\partial y}\right) = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}. $$ That's easy to remember, I think.
It can be made into a rigorous proof by applying the Chain Rule to the composites of $f$ with paths: \begin{gather*} \xi \colon [-\delta, \delta] \to \mathbb{C}, \ t \mapsto (x + t) + iy, \\ \eta \colon [-\delta, \delta] \to \mathbb{C}, \ t \mapsto x + i(y + t), \end{gather*} for small $\delta > 0,$ thus: \begin{multline*} \frac{\partial u}{\partial y} + i\frac{\partial v}{\partial y} = (f \circ \eta)'(0) \\ = f'(\eta(0))\eta'(0) = if'(x + iy) = if'(\xi(0))\xi'(0) \\ = i(f \circ \xi)'(0) = i\left(\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} \right). \end{multline*}