Analytic Functions over Fields other than Real or Complex Numbers Let K denote either the field of real numbers or the complex fields. An analytic function over $K^n$ is a function that can be represented locally by a convergent power series in n variables with coefficients in K.
My question is that can we take K to be other fields? It seems that such a field K should satisfy some criteria:


*

*It is a metric space or at least a topological space.

*There should be complete, in the sense that Cauchy nets converge.

*The above 2 points probably force that K should have cardinality at least the size of the continuum.


Can there be other K where a reasonable theory of analytic functions can be developed? Say for cardinalities larger than the continuum? Probably this invokes some model theory. 
 A: There is a perfectly working theory of analytic functions over the p-adics with lots of theorems. No model theory is needed (Neal Koblitz has a book about that, also Non-Archimedean Analysis by Bosch, Güntzer, Remmert for a dry treatise), but indeed we face (ultra)metric complete fields here, being uncountably infinite, just as you suggest.
Nonetheless, if you just need the "feel" of power series to model something abstractly, formal power series, cf. http://en.wikipedia.org/wiki/Formal_power_series, may be all you need. They behave in many ways like (convergent) power series, for example if you 'formally' wish to invert a differential operator, such computations - at least algebraically - may be given a more-or-less solid foundation in a formal power series ring.
All classical operations, e.g. taking derivatives etc, can be defined termwise, no problem. You can also plug formal power series into each other, but just if the constant coefficient is zero, sadly.
Finally, your two points do not really enforce large cardinality. A finite field can be equipped with the discrete metric, this makes it complete, so you could take about convergent power series over this - it just means that only finitely many coefficients can be non-zero, making it effectively a polynomial ring.
A: If you don´t mind skew fields, have a look at quaternionic analysis. Interestingly, many of the facts of complex analysis don´t hold there at all! (In fact, all you need to do analysis, is a Banach algebra or a complete ring with non-trivial unitary group.)
A: There are several rich theories of analysis on non-archimedian theories.   Neal Koblitz' book on $p$-adic analysis is a good introduction.  Non-archimedian analysis by Bosch, Güntzer and Remmert is more encyclopedic.   Berkovich's Spectral Theory and Analysis over Non-archimedian Fields introduces his beautiful theory of analytic spaces allowing for a reasonable algebraic topological theory.   In Goss's book Basic Structures of Function Field Arithmetic there is a good introduction to analysis in positive characteristic.
Your suggestion that this subject might have something to do with model theory is apt.  As the above references show, the theory may be developed without model theory, but it has been studied intensively via model theory giving interesting results about quantifier elimination, uniformity across the $p$-adics, and establishing a basis for motivic integration.    You might want to look at the paper by van den Dries and Denef,  $p$-adic and real subanalytic sets. Ann. of Math. (2) 128 (1988), no. 1, 79--183.
A: I think that the "right" generalization is "complete valued field" as used in: 
Local Analytic Geometry (Shreeram Shankar Abhyankar) (pag.3).
See: 
http://books.google.it/books?id=h9MmWA4BvD0C&printsec=frontcover&dq=LOCAL+ANALYTIC+GEOMETRY.+by+Shreeram+Shankar+Abhyankar&source=bl&ots=2TM9_FG-u8&sig=UrDked4W1XIPIXXSDcJL9_kuyBI&hl=it&ei=b1YFTOvpOIOAnQO4kI2ODA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false
A: I recommend: Analytic Elements in P-Adic Analysis,  Alain Escassut, World Scientific, 1995.
For other interesting titles see this.
