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.