How to Build Complex Laplace Operator from First Principles

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The discussion centers on the challenge of constructing a Complex Laplace Operator from first principles, specifically using the framework of differential geometry. The original poster expresses uncertainty about where to begin, particularly in applying the exterior derivative to complex-valued functions. They note that while the canonical Laplace operator can be derived from the gradient operator, the application to complex functions remains unclear. Participants suggest considering the standard definitions of derivatives with respect to a complex variable and its conjugate, and exploring the gradient of the divergence of the real and imaginary parts. The conversation highlights a need for clarity on how to effectively build the desired operator in this context.
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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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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).
 
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.
 
I guess you could take the gradient of the divergence of real and imaginary parts of the function on the complex plane?
 

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Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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