Complex exponential: Difference between revisions

Revision as of 21:11, 27 April 2008

Definition

The complex exponential is a holomorphic function from $C$ to $C$ . 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 $\infty$ .

@@ Line 4: / Line 4: @@
 <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>.