note-math

If the exponent is a natural number, then . When , is continuous and monotonically increasing, so the inverse function is also continuous and monotonically increasing, denoted as . can be generalized to rational powers. Due to continuity, can be extended to the power function

Another way is to solve this function if exponential function satisfies the property and

Assume is analytic. Power series expansion of (requires commutativity of ?) (First, assume that there exists a function that satisfies this property, and derive the form , then prove that satisfies this property.)

Expand both sides

Let the coefficients be the same
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[natural-exponential] def with natural-constant

Todo Prove that satisfies . (Techniques used in the proof: absolute convergence, rearrangement invariance, supremum, forced convergence (where one side is supremum), summation in “triangular form” (relative to rectangle) )

from the series, we can see that, differential ==> exists inverse-analytic

[natural-logarithm] def .

for , def

for , def