Definition:Exponential Function/Complex

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Definition

For all definitions of the complex exponential function:

The domain of $\exp$ is $\C$.
The image of $\exp$ is $\C \setminus \left\{ {0}\right\}$, 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 Sum of a Series

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

\(\displaystyle \exp z\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots\) $\quad$ $\quad$


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:

$\displaystyle \exp z := \lim_{n \to \infty} \left({1 + \dfrac z n}\right)^n$


As the Solution of a Differential Equation

The exponential function can be defined as the unique solution $y = f \paren z$ to the first order ODE:

$\dfrac {\d y} {\d z} = y$

satisfying the initial condition $f \paren 0 = 1$.

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 Exponential Function can be found here.