Cauchy-Riemann Equations

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Theorem[edit]

Let be an open subset. Let the function be differentiable at a point . Then all partial derivatives of and exist at and the following Cauchy-Riemann equations hold:

In this case, the derivative of at can be represented by the formula

Proof[edit]

Let . Then

Let . Then

Hence:

Equating the real and imaginary parts, we get the Cauchy-Riemann equations. The representation formula follows from the above line and the Cauchy-Riemann equations.