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 realcomplex 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 realcomplex context as the standard binomial formula does for integer context so does the generalized power rule 

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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 realcomplex 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 realcomplex context as the standard binomial formula does for integer context so does the generalized power rule 

