[real-exponential]
If the exponent is a natural number, then . It can be simply generalized to rational numbers
For the exponent being a real number, define the exponential function as satisfying and
Assume is analytic. Power series expansion of (requires commutativity of ?)
Expand both sides
Let the coefficients be the same
==>
==>
[natural-exponential] def with natural-constant
from the series, we can see that, differential ==> exists inverse-analytic
[natural-logarithm] def .
for , def
for , def
[power-function] Defining the exponential function means that for each , each real exponent is defined, and thus the power function is also defined, or rewritten as
It can also be expressed as
[Euler-formula]