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Is there any known generalization of the riemann mapping theorem over skew-fields of quaternions and beyond or at least a conjectured formulation of it? In a less focused way, how far does the main results of complex analysis and fourier analysis carry over to quaternions.

Koushik
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    Dominic Joyce has a beautiful paper on the subject of quaternionic analysis: Hypercomplex algebraic geometry. Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 194, 129–162. However, there can't be an elementary Riemann mapping theorem due to the rigidity of the possible mappings. – Ben McKay Jan 27 '14 at 13:06
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    Also see Joyce's paper: http://arxiv.org/pdf/math/0010079.pdf – Ben McKay Jan 27 '14 at 13:07
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    You need to be very careful about what you mean by quaternionic analytic mapping. Naively, if you just want a quaternionic Taylor series, there is no nonzero quaternionic-bilinear form to represent the second derivatives, so the mapping becomes linear. There are apparently several ways around this, using more elaborate definitions. – Ben McKay Jan 27 '14 at 18:50
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    Please check the following paper on the subject: C.A. Deavours (1973) "The Quaternion Calculus", American Mathematical Monthly 80:995–1008. – godelian Jul 18 '15 at 13:48
  • This http://mathoverflow.net/questions/51965/is-there-a-quaternionic-algebraic-geometry may be partially of interest for you. – Qfwfq Jul 18 '15 at 16:26
  • and this http://mathoverflow.net/questions/131996/dynamics-in-one-matrix-variable , also slightly related. – Qfwfq Jul 18 '15 at 16:27

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Let $\mathbb{H}$ denote the field of quaternions and $\mathbb{S}$ denote the unit sphere of imaginary quaternions. We have a Riemann mapping theorem for axially symmetric sets inside the field of quaternions $\mathbb{H}$. The set $\Omega$ is axially symmetric if the set $\{ x+Jy~|~J\in\mathbb{S} \},~J\in\mathbb{S}$ are contained in $\Omega$ (we have rotation invariance around the axis). Let $\Omega$ be an axially symmetric open set in $\mathbb{H}$. Extend $\Omega$ to $\mathbb{C}_I$ (the restricted complex plane that can be identified with $\mathbb{C}$) for $I\in\mathbb{S}$. Then if $\mathbb{B}$ is the open unit ball, then we have a bijective mapping of a quaternionic slice function $f$ that bijectively maps from $\Omega$ to $\mathbb{B}$.

Check out the following paper by Sorin Gal and Irene Sabadini: http://arxiv.org/pdf/1406.5516.pdf