I am aware that a quasiconformal map satifies the formula
$$
\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}
$$
where $\sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ imposes a bound on the eccentricity of the ellipses in the image of $f$. For a multicomplex function $F(z\_1, z\_2, \dots, z\_n)$, one could impose the quasiconformality condition for each variable $z\_k$. I have read computer science articles about three-dimensional quasiconformal mappings, but no explicit formula relating the conjugate partial derivative to the partial derivative is given.


**Question:** Does a generalization of the formula $\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}$ apply to quasiconformal mappings in $\mathbb{R}^{2n+1}$?



Quasiconformal maps in arbitrary dimensions
The correct definition of higher-dimensional quasiconformal maps does not use complex variables. The correct condition is
$$
|Df(x)|\le K |J\_f(x)|
$$
where $J\_f$ is the Jacobian determinant. An orientation-preserving homeomorphism $f$ between two domains $U, V$ in $R^n$ is called quasiconformal if it belongs to the Sobolev class $W^{n,1}\_{loc}(U)$ (hence, has 1st order partial derivatives a.e. in $U$) and there exists a constant $K$ such that the above inequality is satisfied a.e. in $U$. There are many other alternative definitions. For instance, instead of working with Sobolev spaces, you can assume that $f$ is absolutely continuous on almost every coordinate line segment in $U$ (hence, has partial derivatives a.e. in $U$) and satisfies the same inequality as above. if you do not know about Sobolev spaces or absolute continuity, just think of $f$ as a diffeomorphism (this is not enough, but suffices for the intuition).


Other definitions are in terms of conformal moduli, conformal capacities, quasisymmetry,...


Some references:


*Iwaniec, Tadeusz; Martin, Gaven*, Geometric function theory and nonlinear analysis, Oxford Mathematical Monographs. Oxford: Oxford University Press (ISBN 0-19-850929-4/hbk). xv, 552 p. (2001). [ZBL1045.30011](https://zbmath.org/?q=an:1045.30011).


*Reshetnyak, Yu. G.*, Space mappings with bounded distortion, Translations of Mathematical Monographs, 73. Providence, RI: American Mathematical Society (AMS). xv, 362 p. (1989). [ZBL0667.30018](https://zbmath.org/?q=an:0667.30018).


*Väisälä, Jussi*, [**Lectures on (n)-dimensional quasiconformal mappings**](http://dx.doi.org/10.1007/BFb0061216), Lecture Notes in Mathematics 229. Berlin-Heidelberg-New York: Springer-Verlag. XIV, 144 p. (1971). [ZBL0221.30031](https://zbmath.org/?q=an:0221.30031).


**Addendum.** The analogue of the Beltrami differential of a map $f$ in higher dimensions is
$$
M\_f(x):= J^{-2n}\_f(x) (Df(x))^T Df(x), 
$$
a field of symmetric positive-definite matrices on $U$. The Beltrami equation
$$
M\_f(x)=A(x),
$$
with $A(x)$ a field of positive-definite symmetric matrices on $U$, is overdetermined if $n\ge 3$ (as Alex noted in his answer). Nevertheless, this equation is sometimes useful when working with quasiconformal maps, although not as useful as the classical Beltrami equation: Most tools in higher dimensions are not analytic but geometric. An interesting thing is that if $A(x)$ is smooth, then there are known necessary and sufficient conditions for solvability of the higher-dimensional Beltrami equation (the condition is a 3rd order nonlinear PDE on $A$ if $n=3$ and a 2nd order nonlinear PDE on $A$ if $n\ge 4$). I do not know if anybody worked out a distributional analogue of this classical result.


