Definition:Exponential Function/Complex/Differential Equation

Definition
Let $\exp: \C \to \C \setminus \left\{ {0}\right\}$ denote the (complex) exponential function. The exponential function can be defined as the unique solution $y = f \left({z}\right)$ to the first order ODE:


 * $\dfrac {\mathrm d y} {\mathrm d z} = y$

satisfying the initial condition $f \left({0}\right) = 1$.

That is, the defining property of $\exp$ is that it is its own derivative.

The complex number $\exp z$ is called the exponential of $z$.