What are the Cauchy-Riemann equations and their geometric interpretation?

arshavin
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Is there a geometric meaning for the derivative of a complex valued function, or any other motivation for the derivative?
 
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Do you mean the Cauchy-Riemann equations? They just say that if you calculate the directional derivative along any line through a point you'll get the same answer regardless of direction. Write down the directional derivative of a function along the real direction and the imaginary direction and set them equal. You'll get CR.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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