# Analytic Functions - why 2 dimensions?

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?

• 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.
– mrf
Jan 13, 2017 at 11:41
• In particular, they must be harmonic conjugates of each other. That is, their gradients are perpendicular everywhere. Jan 13, 2017 at 15:34

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 $$\operatorname{div}u=0,\quad \operatorname{curl}u=0.$$
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.