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Having just learned the definition of analytic functions, I found it surprising and somewhat counterintuitive that the set of functions over the complex plane whose real and imaginary parts each satisfy Laplace's Equation $\nabla^{2} = 0$ should have such incredible significance, since this is basically a set of two harmonic functions over $\mathbb{R}$. Does this extend to higher dimensions of hypercomplex numbers such as quaternions?

More formally, is there some generalization of the space of analytic functions to higher dimensions $ 2^n | n \in \mathbb{N}, n > 1$ where each component function individually satisfies Laplace's Equation? Do these functional spaces yield similar properties to that enumerated in the field of complex analysis, or are there any salient differences?

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    $\begingroup$ A holomorphic (analytic) function is not just a pair of harmonic functions. They have to have another interrelation too, captured by Cauchy-Riemann's equations. $\endgroup$ – mrf Jan 13 '17 at 11:41
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    $\begingroup$ In particular, they must be harmonic conjugates of each other. That is, their gradients are perpendicular everywhere. $\endgroup$ – eyeballfrog Jan 13 '17 at 15:34
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In the book Introduction to Fourier Analysis on Euclidean Spaces, by E.M. Stein and G. Weiss, a "generalized analytic function" is defined as a function $u=(u_1,\dots,u_n)\colon D\subset\mathbb{R}^n\to\mathbb{R}^n$ such that $$ \sum_{i=1}^n\frac{\partial u_i}{\partial x_i}=0,\quad \frac{\partial u_i}{\partial x_j}=\frac{\partial u_j}{\partial x_i},\quad1\le i,j\le n. $$ This is equivalent to $$ \operatorname{div}u=0,\quad \operatorname{curl}u=0. $$

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In quaternion analysis the notions of analyticity, holomorphy, and harmonicity are not linked as for complex analysis as you can see in this presentation of quaternionic analysis whre it's presented the definition of a ''regular'' quaternionic function proposed by R. Fueter ( in 1935), characterized by a property analogue to the Cauchy-Riemann equations. Obviously many difficults in the extension of calculus to quaternions come from non commutativity, that is a characteristic of any extension of the field of complex numbers.

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It is also amazing that domains in $\mathbb R^2$ have so many conformal maps, but domains in $\mathbb R^n$ for $n \ge 3$ have (by comparison) hardly any.

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