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| Title of object: |
complex exponential function |
| Canonical Name: |
ComplexExponentialFunction |
| Type: |
Definition |
| Created on: |
2004-10-10 03:21:32 |
| Modified on: |
2005-05-10 13:40:31 |
| Classification: |
msc:30D20, msc:32A05 |
| Defines: |
exponential function, prime period |
Preamble:
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Content:
The {\em complex exponential function} \,\,$\exp:\,\mathbb{C}\to \mathbb{C}$\, may be defined in many equivalent ways: \,Let \,$z = x+iy$\, where \,$x,\,y\in\mathbb{R}$.
\begin{itemize}
\item $\exp{z} := e^x(\cos{y}+i\sin{y})$
\item $\exp{z} := \lim_{n\to\infty}(1+\frac{z}{n})^n$
\item $\exp{z} := \sum_{n = 0}^\infty\frac{z^n}{n!}$
\end{itemize}
The complex exponential function is usually denoted in power form:
$$e^z := \exp{z},$$
where $e$ is the Euler number. \,It also coincincides with the real exponential function when $z$ is real (choose \,$y = 0$). \,It has all the properties of power, e.g. \,$e^{-z} = \frac{1}{e^z}$; \,these are consequences of the addition formula
$$e^{z_1+z_2} = e^{z_1}e^{z_2}$$
of the complex exponential function.
The function gets all complex values except 0 and is periodic having the \PMlinkescapetext{{\em prime period}} (the \PMlinkescapetext{period} with least non-zero modulus) $2\pi i$. \,The $\exp$ is holomorphic, its derivative
$$\frac{d}{dz}e^z = e^z,$$
which is obtained from the series form, is similar as in $\mathbb{R}$. \,
Some formulae:
$$|e^z| = e^x, \quad \arg{e^z} = y+2n\pi\,\,\,(n = 0,\,\pm1,\,\pm2,\,...),$$
$$\mbox{Re}(e^z) = e^x\cos{y}, \quad \mbox{Im}(e^z) = e^x\sin{y}$$ |
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