Cauchy-Riemann Equations

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

The 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

edit

Let  . Then

 

Proof - Step 2

edit

Let  . Then

 

Proof - Step 3

edit

Both partial derivatives must provide a same complex value due to the fact that   is complex differentiable:  

Proof - Step 4

edit

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.

See also

edit


Page Information

edit

You can display this page as Wiki2Reveal slides

Wiki2Reveal

edit

The Wiki2Reveal slides were created for the Complex Analysis' and the Link for the Wiki2Reveal Slides was created with the link generator.