# Properties of Complex Exponential Function

## Theorem

Let $z \in \C$ be a complex number.

Let $\exp z$ be the exponential of $z$.

Then:

### Image of Complex Exponential Function

The image of the complex exponential function is $\C \setminus \left\{ {0}\right\}$.

### Exponential of Sum

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\exp z$ be the exponential of $z$.

Then:

$\map \exp {z_1 + z_2} = \paren {\exp z_1} \paren {\exp z_2}$

### Reciprocal of Complex Exponential

$\dfrac 1 {\map \exp z} = \map \exp {-z}$

### Complex Exponential Function has Imaginary Period

$\map \exp {z + 2 k \pi i} = \map \exp z$

### Exponential Function is Continuous

$\forall z_0 \in \C: \displaystyle \lim_{z \mathop \to z_0} \exp z = \exp z_0$

### Derivative of Exponential Function

The complex exponential function is its own derivative.

That is:

$\map {D_z} {\exp z} = \exp z$