# Definition:Exponential Function/Complex/Differential Equation

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## Definition

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 $z$**.