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