Complex exponential: Difference between revisions

Latest revision as of 19:11, 18 May 2008

This article is about a particular entire function: a holomorphic function defined on the whole of
$C$

View a complete list of entire functions

Definition

The complex exponential is a holomorphic function from $C$ to $C$ (i.e. an entire function). The complex exponential of $z$ is denoted as $\exp (z)$ and also as $e^{z}$ . It is defined as the sum of the following power series:

$\exp (z) = \sum_{n = 0}^{\infty} \frac{z^{n}}{n!}$

Alternatively, it can be defined in terms of the real exponential, real sine function, and real cosine function, as follows:

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$

The complex exponential does not extend to a meromorphic function on the Riemann sphere: it has an essential singularity at $\infty$ .

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+{{particular entire function}}
 ==Definition==
-The '''complex exponential''' is a [[holomorphic function]] from <math>\mathbb{C}</math> to <math>\mathbb{C}</math>. The complex exponential of <math>z</math> is denoted as <math>\exp(z)</math> and also as <math>e^z</math>. It is defined as the sum of the following power series:
+The '''complex exponential''' is a [[holomorphic function]] from <math>\mathbb{C}</math> to <math>\mathbb{C}</math> (i.e. an [[entire function]]). The complex exponential of <math>z</math> is denoted as <math>\exp(z)</math> and also as <math>e^z</math>. It is defined as the sum of the following power series:
 <math>\exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!}</math>
+Alternatively, it can be defined in terms of the [[real exponential]], [[real sine function]], and [[real cosine function]], as follows:
+<math>\exp(x + iy) = e^x(\cos y + i \sin y)</math>
 The complex exponential does ''not'' extend to a meromorphic function on the [[Riemann sphere]]: it has an [[essential singularity]] at <math>\infty</math>.