The complex exponential function may be defined in many equivalent ways: Let where .
The complex exponential function is usually denoted in power form:
where is the Napier’s constant. It also coincincides with the real exponential function when is real (choose ). It has all the properties of power, e.g. ; these are consequences of the addition formula
of the complex exponential function.
The function gets all complex values except 0 and is periodic having the prime period (the period with least non-zero modulus) . The is holomorphic, its derivative
which is obtained from the series form via termwise differentiation, is similar as in .
So we have a fourth way to define
with the solution of the differential equation under the initial condition .