How to Build Complex Laplace Operator from First Principles

In summary, The conversation discusses the concept of a Complex Laplace Operator and its construction from first principles in differential geometry. The canonical Laplace operator or Laplace matrix is typically used for continuous or discrete functions, but the question arises for complex-valued functions. It is suggested that taking the gradient of the divergence of the real and imaginary parts of the function on the complex plane could potentially lead to the desired Hodge Laplacian. However, there is uncertainty on how to use this approach.
  • #1
topcomer
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Has anyone ever heard about a Complex Laplace Operator? I would like to build one from first principles as in differential geometry ∆=d*d, where d is the exterior derivative, but I don't know where to start. Actually, I was even unsure in which forum to post the question.

If one defines d to be the gradient operator acting on continuous or discrete functions, then one gets the canonical Laplace operator or the Laplace matrix used in graph theory and image processing. But what if one consider complex-valued functions?
 
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  • #2
Has this been done? For complex function w(z) = f(x) + i.g(y), z = x + i.y... I don't see how to get a Laplace operator from it! Would be interested to know how it is done though, so I think I'll subscribe (and bump).
 
  • #3
MikeyW said:
Has this been done? For complex function w(z) = f(x) + i.g(y), z = x + i.y... I don't see how to get a Laplace operator from it! Would be interested to know how it is done though, so I think I'll subscribe (and bump).

I have no idea, for complex functions there is a standard definition of taking the derivative w.r.t. z and its conjugate, but I don't see how to use this to construct the Hodge Laplacian I'm interested in.
 
  • #4
I guess you could take the gradient of the divergence of real and imaginary parts of the function on the complex plane?
 

1. What is the Laplace operator?

The Laplace operator, also known as the Laplacian, is a mathematical operator that is used in many branches of science and engineering to describe the rate of change of a physical quantity with respect to its spatial coordinates.

2. Why is it important to build a complex Laplace operator from first principles?

Building a complex Laplace operator from first principles allows for a more thorough understanding of how it operates and how it can be applied in different contexts. It also lays the foundation for further development and customization of the operator for specific applications.

3. What are the first principles used to build the complex Laplace operator?

The first principles used to build the complex Laplace operator include the fundamental properties of the Laplacian, such as linearity and symmetry, as well as basic concepts from vector calculus, such as the gradient and divergence operators.

4. How is the complex Laplace operator used in practice?

The complex Laplace operator is commonly used in fields such as physics, engineering, and mathematics to solve differential equations that describe physical systems. It is also used in image and signal processing, as well as in machine learning algorithms.

5. Are there any challenges in building a complex Laplace operator from first principles?

Building a complex Laplace operator from first principles can be challenging due to its abstract nature and the complex mathematical concepts involved. It requires a strong understanding of vector calculus, linear algebra, and complex analysis, as well as the ability to apply these concepts in a systematic and logical way.

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