Cauchy-Riemann Equations
Theorem
editLet 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
editThe proof can be decomposed into 3 main steps:
- calculate the partial derivative for the real part,
- calculate the partial derivative for the imaginary part,
- due to property of being complex differentiable both derivatives yield the same complex value. This leads to Cauchy-Riemann equations.
Proof - Step 1
editLet . Then
Proof - Step 2
editLet . Then
Proof - Step 3
editBoth partial derivatives must provide a same complex value due to the fact that is complex differentiable:
Proof - Step 4
editEquating the real and imaginary parts, we get the Cauchy-Riemann equations. The representation formula follows from the above line and the Cauchy-Riemann equations.
See also
edit
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