# 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

• Results about the exponential function can be found here.