Hi, I have a question about Cauchy Reimann equation(adsbygoogle = window.adsbygoogle || []).push({});

lets say

z=x+yi is in R^2

And there exists

f:R^2->R^2

f(z)=u(x,y)+v(x,y)i

Then cauchy reimann condition states that

If partial x of f and y of f are equal, then f is holomorphic

However, im not sure how this can be a necesary sufficient condition because it only considers y and x derivatives but does not consider directional derivatives.

Because f (z) is essentially the same as a vector field, its component u (x,y) and v (x,y) will have directional derivatives at a given point p.

And by the definition of holomorphicity, it should have the same derivative regardless of direction of derivatives.

Could anyone explain me how the cuachy reimann condition implies this?

THANKS

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# I Cauchy Reimann Condition

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