[goldenkey]

Complex conjugation is non-holomorphic and yes, it's ugly. I still don't see why it being useful as a construction makes it a beautiful operation in analysis given that it isn't algebra.

There are holomorphic split-complex functions and they aren't uninteresting at all - they are quite beautiful and complex, relating to wave equations: https://en.wikipedia.org/wiki/Motor_variable#D-holomorphic_f...

They're not something often looked into in the fractal scene but if anyone has spare time, I'd love to see a D-holomorphic fractal analog of the Mandelbrot :-)

[ogogmad]

A holomorphic function over the split-complex numbers is the direct sum of two holomorphic functions over R. In that sense, it's trivial.

You will be very disappointed once you see the split-complex Mandelbrot set. It will be an instructive exercise for you to see why it looks the way it does.

My point about complex conjugation is illustrated in [1]. Linear algebra over the complex numbers does not limit itself to the four arithmetic operations {+,-,*,/}, but uses complex conjugation as the 5th and final operation. Look at the definition of a unitary matrix, or a self-adjoint matrix, or a normal matrix. Also, look at the definitions of the QR decomposition and SVD. Complex conjugation is everywhere in linear algebra. Consequently, matrix theory over the split-complex numbers is actually somewhat interesting.

[1] - https://en.wikipedia.org/wiki/Matrix_decomposition

To some extent, this seems like a weird discussion about mathematical aesthetics.

[goldenkey]

Linear algebra uses the Hermitian adjoint [1], sure, but how often do we graph or build functions out of it for analysis purposes? I understand that it is useful in formulating equations, relations, etc.

Linear algebra does a lot of things, ie. transpose, that aren't algebraic. I think it's important we understand what is computational versus algebraic. In that vain, linear algebra proves to be immensely useful, but we will trip ourselves if we start thinking certain aspects of it are analytic.

Anyhow, point taken, that image hardly looks interesting.

I think we agree and like you said, are just talking about aesthetics. I agree that the utility of math doesn't end at algebra or analyticity.

https://en.wikipedia.org/wiki/Hermitian_adjoint

[ogogmad]

Here's the split-complex Mandelbrot (and Mandelbar for comparison): https://imgur.com/a/j2c7qkB

Read my other reply.

[goldenkey]

Thank you.