# Definition:Exponential Function/Complex

## Definition

For all definitions of the **complex exponential function**:

- The domain of $\exp$ is $\C$.

- The image of $\exp$ is $\C \setminus \set 0$, as shown in Image of Complex Exponential Function.

For $z \in \C$, the complex number $\exp z$ is called the **exponential of $z$**.

### As a Power Series Expansion

The **exponential function** can be defined as a (complex) power series:

\(\ds \forall z \in \C: \, \) | \(\ds \exp z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots\) |

### By Real Functions

The **exponential function** can be defined by the real exponential, sine and cosine functions:

- $\exp z := e^x \paren {\cos y + i \sin y}$

where $z = x + i y$ with $x, y \in \R$.

Here, $e^x$ denotes the real exponential function, which must be defined first.

### As a Limit of a Sequence

The **exponential function** can be defined as a limit of a sequence:

- $\ds \exp z := \lim_{n \mathop \to \infty} \paren {1 + \dfrac z n}^n$

### As the Solution of a Differential Equation

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.

## Notation

The **exponential of $z$**, $\exp z$, is frequently written as $e^z$, as in the case of the real exponential.

## Also see

- Equivalence of Definitions of Complex Exponential Function
- Properties of Complex Exponential Function

- Results about
**the exponential function**can be found**here**.