Rick's Random Interesting Things in Maths

Quaternion Cauchy's Theorem

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. Surely quaternionic analytic functions should be richer still. Lamentably the dream remains just that.

One of the key attributes which makes analytic functions of a complex variable so useful is Cauchy's theorem. Is there a quaternionic analogue of Cauchy's theorem? Yes there is. It is presented here.

Quaternion Analogue of Cauchy's Theorem

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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.