How to Build Complex Laplace Operator from First Principles

[topcomer]

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?

 

[mikeph]

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).

 

[topcomer]

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.

 

[mikeph]

I guess you could take the gradient of the divergence of real and imaginary parts of the function on the complex plane?