Proof of Harmonic Function Infinitely Differentiable

In summary, the conversation discusses the concept of a harmonic function and its relation to being infinitely differentiable. The proof presented in the book involves using a theorem and a disk in a simply-connected region. The question is raised about the applicability of this proof for points on the boundary of the region. The answer is that typically, these points are not considered and strict conditions would need to be met for them to be included. Additionally, the region G is assumed to be open and connected.
  • #1
Silviu
624
11
Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that ##f = u + iv## is holomorphic in G. ". In the proof they present in the book they begin with: "Suppose u is harmonic in G and ##z_0 ∈ G##. Let ##r > 0## such that the disk ##D[z_0, r]## is contained in G. " and as a disk is simply connected the conclusion follows from the theorem. My question is, how can you make sure that for any ##z_0 \in G## you can have a disk around ##z_0##? (For example if ##G=\mathbb{C}##\##\mathbb{R}_{<0}## and ##z_0=0##, you can't find such a disk. What am I missing?
 
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  • #2
Typically, these things are only considered in the interior of the set G. Statements about points on the boundary of G would require a lot of conditions and restrictions.
 
  • #3
they evidently assume ##G## to be an open set:
Silviu said:
Let r>0r > 0 such that the disk D[z0,r]D[z_0, r] is contained in G.
 
  • #4
Region usually means open and connected.
 
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  • #5
Silviu said:
For example if G=CG=\mathbb{C}\R<0\mathbb{R}_{z0=0z_0=0, you can't find such a disk.
by the way, what is a harmonic function at such a point ##z_0##?
 

Related to Proof of Harmonic Function Infinitely Differentiable

Question 1: What is a proof of harmonic function infinitely differentiable?

A proof of harmonic function infinitely differentiable is a mathematical demonstration that shows a function, which satisfies Laplace's equation, is infinitely differentiable. This means that the function has derivatives of all orders at every point within its domain.

Question 2: Why is it important to prove that a harmonic function is infinitely differentiable?

This proof is important because it ensures that the function is smooth and well-behaved, making it easier to analyze and use in various mathematical and scientific applications. Additionally, it provides a deeper understanding of the behavior of harmonic functions.

Question 3: How is the proof of harmonic function infinitely differentiable typically conducted?

The proof is typically conducted using complex analysis techniques, such as Cauchy-Riemann equations and the Cauchy integral formula. These tools allow for a rigorous and systematic approach to demonstrating the infinite differentiability of harmonic functions.

Question 4: What are some applications of the proof of harmonic function infinitely differentiable?

The proof has various applications in physics, engineering, and other scientific fields. It is used to analyze and solve problems involving electric fields, fluid flow, and heat transfer, among others. It also has applications in signal processing and image reconstruction.

Question 5: Are there any limitations to the proof of harmonic function infinitely differentiable?

While the proof is valid for functions that satisfy Laplace's equation, it may not be applicable to functions that do not meet this criteria. Additionally, the proof may become more complex and challenging for functions with more complex domains or boundary conditions.

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