Definition:Exponential Function/Complex/Differential Equation

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 $z$.

Also see

• Equivalence of Definitions of Complex Exponential Function

• Results about the exponential function can be found here.