Let F be such a real closed field and let F(C) be the algebraic closure of F. An important example of F is FSR- the field of SURREAL numbers. Suppose that one has constructed an exponential function f which maps F into itself and which is an extension of the exponential function already defined on the proper subfield of F that is (isomorphic to) the field of real numbers. H. Gonshor has done this for FSR. My question is:
Is there a natural way to extend f to f* so that f* is a mapping of F(C) into itself which is also an extension of the already defined exponential function on the proper subfield of F(C) that is (isomorphic to) the field of complex numbers?
