\begin{definition}[Definition:Exponential Function/Complex/Differential Equation]
Let $\exp: \C \to \C \setminus \set 0$ denote the '''(complex) exponential function'''.
The '''exponential function''' can be defined as the unique particular solution $y = \map f z$ to the first order ODE:
:$\dfrac {\d y} {\d z} = y$
satisfying the initial condition $\map f 0 = 1$.
That is, the defining property of $\exp$ is that it is its own derivative.
The complex number $\exp z$ is called the '''exponential of $z$'''.
\end{definition}