Complex differentiability problem

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SUMMARY

The discussion centers on the concept of holomorphic functions and their differentiability. It is established that holomorphic functions, which satisfy the Cauchy-Riemann (C-R) equations, are indeed infinitely differentiable, regardless of whether their derivatives are non-zero. The example provided, where z = x + iy has a first derivative of 1 and subsequent derivatives of 0, illustrates that a function can be infinitely differentiable while having higher derivatives equal to zero.

PREREQUISITES
  • Understanding of holomorphic functions
  • Familiarity with Cauchy-Riemann equations
  • Basic knowledge of complex differentiation
  • Concept of infinite differentiability
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  • Study the implications of the Cauchy-Riemann equations in complex analysis
  • Explore the properties of holomorphic functions in detail
  • Learn about Taylor series expansions for complex functions
  • Investigate examples of functions that are infinitely differentiable but have zero derivatives beyond the first
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Students and professionals in mathematics, particularly those studying complex analysis, as well as educators looking to clarify concepts related to holomorphic functions and differentiability.

Epsilon36819
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Hi guys,

I have this problem understanding that holomorpic functions must be infinitely differentiable. Indeed, it does follow from the Cauchy formula. But take z=x+iy. It satisfies C-R equations and has a first derivative = 1. I fail to see how this function is infinitely differentiable.

What am I missing?

Thanks!
 
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Epsilon36819 said:
But take z=x+iy. It satisfies C-R equations and has a first derivative = 1.

So its second derivative is 0.

And its third, and its fourth, and …

Being (infinitely) differentiable simply means that it has a derivative - the derivative doesn't have to be non-zero.

It's like, is zero a number? You could say it's not anything, so it's not a number … but it is! :smile:
 
That does make sense. I really did read this as "a non zero" derivative...:redface:

Thanks for clearing that up! :smile:
 

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