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Version 17 |
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The {\em complex exponential function}\,\, $\exp:\,\mathbb{C}\to \mathbb{C}$\, may be defined in many equivalent ways:\, Let\, $z = x\!+\!iy$\, where\, $x,\,y\in\mathbb{R}$.
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The {\em complex exponential function}\,\, $\exp:\,\mathbb{C}\to \mathbb{C}$\, may be defined in many equivalent ways:\, Let\, $z = x+iy$\, where\, $x,\,y\in\mathbb{R}$.
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| \begin{itemize} |
\begin{itemize} |
| \item $\displaystyle\exp{z} \;:=\; e^x(\cos{y}+i\sin{y})$ |
\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 $\displaystyle\exp{z} \;:=\; \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n$ |
| \item $\displaystyle\exp{z} \;:=\; \sum_{n = 0}^\infty\frac{z^n}{n!}$ |
\item $\displaystyle\exp{z} \;:=\; \sum_{n = 0}^\infty\frac{z^n}{n!}$ |
| \end{itemize} |
\end{itemize} |
| The complex exponential function is usually denoted in power form: |
The complex exponential function is usually denoted in power form: |
| $$e^z \;:=\; \exp{z},$$ |
$$e^z \;:=\; \exp{z},$$ |
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where $e$ is the Napier's constant.\, 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
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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
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| $$e^{z_1+z_2} \;=\; e^{z_1}e^{z_2}$$ |
$$e^{z_1+z_2} \;=\; e^{z_1}e^{z_2}$$ |
| of the complex exponential function. |
of the complex exponential function. |
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| 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 |
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 |
| $$\frac{d}{dz}e^z \;=\; e^z,$$ |
$$\frac{d}{dz}e^z \;=\; e^z,$$ |
| which is obtained from the series form via termwise differentiation, is similar as in $\mathbb{R}$. |
which is obtained from the series form via termwise differentiation, is similar as in $\mathbb{R}$. |
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| So we have a fourth way to define |
So we have a fourth way to define |
| \begin{itemize} |
\begin{itemize} |
| \item $\exp{z} \;:=\; w(z)$ |
\item $\exp{z} \;:=\; w(z)$ |
| \end{itemize} |
\end{itemize} |
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with $w$ the solution of the differential equation \,$\displaystyle\frac{dw}{dz} = w$\, under the initial condition\, $w(0) = 1$.
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with $w$ the solution of the differential equation\, $\displaystyle\frac{dw}{dz} = w$\, under the initial condition\, $w(0) = 1$.
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| \textbf{Some formulae:} |
\textbf{Some formulae:} |
| $$|e^z| \;=\; e^x, \quad \arg{e^z} \;=\; y+2n\pi\quad(n = 0,\,\pm1,\,\pm2,\,\ldots),$$ |
$$|e^z| \;=\; e^x, \quad \arg{e^z} \;=\; y+2n\pi\quad(n = 0,\,\pm1,\,\pm2,\,\ldots),$$ |
| $$\mbox{Re}(e^z) \;=\; e^x\cos{y}, \quad \mbox{Im}(e^z) \;=\; e^x\sin{y}$$ |
$$\mbox{Re}(e^z) \;=\; e^x\cos{y}, \quad \mbox{Im}(e^z) \;=\; e^x\sin{y}$$ |
