Rick's Random Interesting Things in Maths

Cauchy-type Integral Theorems for Quaternion Functions

Quaternions are a will'o the wisp. Many people have been lost in their mathematical marshes. The perennial dream is to discover how to make quaternions even more powerful than complex numbers. Mathematical analysis is virtually identified with functions of a complex variable, especially the holomorphic (or analytic) functions. Surely quaternionic analytic functions should be richer still. There is every possibility they could be, though that potential has not yet been exploited.

One of the key attributes which makes holomorphic functions of a complex variable so useful is Cauchy's theorem. Is there a quaternionic analogue of Cauchy's theorem? Yes there is. The first of these was not known to Hamilton and the other nineteenth century mathematicians but was discovered only in 1935 by the German, Fueter. Some twenty years or so ago I discovered an alternative Cauchy-type integral theorem but I believed it was not original - and, indeed, that it was a mere recasting of Fueter's Theorem. This web site has been making that assertion for many years. However, I am now of the view that the two theorems are distinct as their key feature - independence of the integration surface - arises from different properties - and I can see no way of directly derving one theorem from the other.

In the first linked pdf below, the "paper", I present the most elegant and succinct derivation of both theorems, emphasising whence they originate. This crucially depoloys the Stokes-Cartan theorem and the concepts of exterior algebra. The second linked pdf is based on my original notes from about twenty years ago and is far more detailed, far longer and less elegant - but does not require the Stokes-Cartan theorem or familiarity with exterior algebra.

Both pdfs also present the theorems for the case of biquaternions which provide a link with Minkowski spacetime and an elegant formulation of the Lorentz transform.

RAWB, 28 May 2023.

Cauchy-type Integral Theorems for Quaternion and Biquaternion Functions: The Paper

Cauchy-type Integral Theorems for Quaternion and Biquaternion Functions: My Original Version circa 2003

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The familiar 2D Mandlebrot set can be made 3D with a little ingenuity. The above are a couple of illustrations of what results. Details can be found here, whence these images derive.