## complex curvilinear coordinates(?)

Hello,
a system of curvilinear coordinates is usually expressed by an admissible transformation represented by a set of real scalar functions $x_i=x_i(u_1,\ldots,u_n)$.

Does it make sense to form a system of curvilinear coordinates where the [itex]x_i[/tex] and [itex]u_i[/tex] functions are complex valued?
If that exists, I cannot find any source in the net.
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 Wouldn't that be equivalent to a 2n coordinates? But I'm not quite sure
 They are called "Complex manifolds"; imporatnt examples in Physics are the Calabi-Yau manifolds and Riemann surfaces, see: http://en.wikipedia.org/wiki/Complex_manifold