Cauchy-Riemann Equations

The partial derivative of an analytic function is the same in any direction. In particular, The partial derivative along the real axis is the same as along the imaginary axis. Therefore, we have.

\begin{align} \renewcommand\d{\mathrm{d}} \newcommand{\p}[2]{\frac{\partial#1}{\partial#2}} \p{u}{x}+i\p{v}{x}=\p{f}{x}=\frac{1}{i}\p{f}{y}=-i\cdot(\p{u}{y}+i\p{v}{y})=\p{v}{y}-i\p{u}{y}. \end{align}

It follows that

\begin{align} \p{u}{x}=\p{v}{y}\qquad\p{v}{x}=-\p{u}{y}, \end{align}

which are the so-called Cauchy-Riemann Equations

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