# 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 $y = f \paren z$ to the first order ODE:

- $\dfrac {\d y} {\d z} = y$

satisfying the initial condition $f \paren 0 = 1$.

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

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