The Exponential Function
The Exponential Function:
We define the complex exponential as follows:
1. If z = x is real, then
as in calculus. Here ! denotes “factorial”: j! = j·(j−1)·(j−2) · · · 3·2·1.
2. If z = iy is pure imaginary, then
3. If z = x iy, then
Part and parcel of the last definition of the exponential is the following complex-analytic definition of the sine and cosine functions:
Note that when z = x i0 is real this new definition coincides with the familiar Euler formula from calculus:
Exponential Using Power Series:
It is also possible to define the exponential using power series:
Either definition is correct for any z, and they are logically equivalent.
Laws of Exponentiation:
The complex exponential satisfies familiar rules of exponentiation:
One may verify these properties directly from the power series definition, or else use the more explicit definitions in the above equations.