Complex differentiability problem

In summary, the conversation discusses the concept of holomorphic functions being infinitely differentiable. The speaker has difficulty understanding how this is possible, but it is explained that the function only needs to have a derivative, regardless of its value, to be considered differentiable. The conversation ends with the speaker understanding this concept.
  • #1
Epsilon36819
32
0
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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  • #2
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:
 
  • #3
That does make sense. I really did read this as "a non zero" derivative...:redface:

Thanks for clearing that up! :smile:
 

1. What is complex differentiability?

Complex differentiability refers to the property of a function to have a derivative at a particular point in the complex plane. This means that the function can be approximated by a linear function at that point.

2. How is complex differentiability different from real differentiability?

In real differentiability, a function is differentiable if it has a well-defined tangent line at a point. However, in complex differentiability, a function must have a well-defined tangent line in both the real and imaginary directions at a point, which is known as the Cauchy-Riemann equations.

3. Why is complex differentiability important?

Complex differentiability is important in complex analysis, a branch of mathematics that studies functions of complex variables. It allows for the use of calculus techniques to analyze complex functions, which has a wide range of applications in physics, engineering, and other fields.

4. What are the conditions for a function to be complex differentiable?

A function f(z) is complex differentiable at a point z0 if and only if it satisfies the Cauchy-Riemann equations at that point, which state that the partial derivatives of f with respect to x and y must exist and be continuous at z0.

5. Can a function be complex differentiable at some points but not others?

Yes, a function can be complex differentiable at some points in the complex plane but not others. This is because the Cauchy-Riemann equations must be satisfied at each individual point for a function to be complex differentiable at that point.

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