complex exponential functionComplexExponentialFunction2004-10-10 03:21:322009-09-28 17:45:16Definitionexponential functionexponential functionprime period% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe {\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 $\displaystyle\exp{z} \;:=\; e^x(\cos{y}+i\sin{y})$
\item $\displaystyle\exp{z} \;:=\; \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n$
\item $\displaystyle\exp{z} \;:=\; \sum_{n = 0}^\infty\frac{z^n}{n!}$
\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 \PMlinkname{periodic}{PeriodicityOfExponentialFunction} 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 via termwise differentiation, is similar as in $\mathbb{R}$.
So we have a fourth way to define
\begin{itemize}
\item $\exp{z} \;:=\; w(z)$
\end{itemize}
with $w$ the solution of the differential equation \,$\displaystyle\frac{dw}{dz} = w$\, under the initial condition\, $w(0) = 1$.
\textbf{Some formulae:}
$$|e^z| \;=\; e^x, \quad \arg{e^z} \;=\; y+2n\pi\quad(n = 0,\,\pm1,\,\pm2,\,\ldots),$$
$$\mbox{Re}(e^z) \;=\; e^x\cos{y}, \quad \mbox{Im}(e^z) \;=\; e^x\sin{y}$$