You have presented no sources to support your apparently original idea that  has a different definition depending on whether the exponent has “integer context” or “real/complex context”. If you find such a source, you are welcome to add a third viewpoint, but this is not a distinction made by Knuth, even though he happens to use integers in his argument. (They are not fundamental to his argument. The generalized binomial theorem makes just as convincing a case for  for an exponent in “real/complex context” as the standard binomial formula does for “integer context”; so does the generalized power rule.) —