Is this function holomorphic?

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In summary, the function f(z) = x^2 + iy^2 is not analytic at any point, as it does not satisfy the Cauchy-Riemann equations in any neighborhood. The term "holomorphic anywhere" does not make sense, as holomorphic means analytic for all complex numbers. It would be more accurate to ask if the function is analytic anywhere.
  • #1
blahblah8724
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If [itex] z = x + iy [/itex] then the function [itex] f(z) := x^2 + iy^2 [/itex], has total derivative,

[itex] \begin{pmatrix} 2x & 0 \\ 0 & 2y \end{pmatrix} [/itex]

so surely by the Cauchy–Riemann equations this is complex differentiable at [itex] x = y [/itex], but is this function holomorphic anywhere?

Thanks!
 
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No, it is not analytic anywhere. In order to be analytic at a point, the Cauchy-Riemann equations must be satisfied in some neighborhood of the point and that does not happen here.

By the way, the phrase "holomorphic anywhere" does not make sense. "Holomorphic" means "analytic for all complex numbers". You mean to ask if it was analytic anywhere.
 
  • #3
HallsofIvy said:
By the way, the phrase "holomorphic anywhere" does not make sense. "Holomorphic" means "analytic for all complex numbers". You mean to ask if it was analytic anywhere.
? According to the Wikipedia article, holomorphic at a point means complex-differentiable in some neighborhood of the point. So asking whether a function is holomorphic anywhere does make sense.
 

1. What does it mean for a function to be holomorphic?

A function is said to be holomorphic if it is complex differentiable at every point within its domain. This means that the function has a well-defined derivative at each point and can be represented as a power series.

2. Can a real-valued function be holomorphic?

No, a real-valued function cannot be holomorphic because it does not satisfy the Cauchy-Riemann equations, which are necessary conditions for a function to be holomorphic.

3. How can I determine if a function is holomorphic?

To determine if a function is holomorphic, you can check if it satisfies the Cauchy-Riemann equations, which state that the partial derivatives of the function with respect to x and y must exist and be continuous, and that they must satisfy a specific relationship. You can also check if the function is complex differentiable at every point within its domain.

4. What is the relationship between holomorphic functions and analyticity?

A function that is holomorphic is also analytic, meaning it can be represented as a convergent power series. However, not all analytic functions are holomorphic. For a function to be holomorphic, it must satisfy additional conditions such as the Cauchy-Riemann equations.

5. Are there any real-world applications of holomorphic functions?

Yes, holomorphic functions have many real-world applications, particularly in the fields of physics and engineering. They are used to model and describe physical systems, such as fluid flow and electromagnetism, and are also used in signal processing and image reconstruction.

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