complex logarithmComplexLogarithm2004-10-10 03:46:212008-01-25 07:51:07Definitioncomplex exponential function% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe \PMlinkescapetext{{\em logarithm} of a complex number} $z$ is defined as every complex number $w$ which satisfies the equation
\begin{align}
e^w = z.
\end{align}
This is is denoted by
$$\log{z} := w.$$
The solution of (1) is obtained by using the form \,$e^w = re^{i\varphi}$\,, where\, $r = |z|$\, and \,$\varphi = \arg{z}$;\, the result is
$$w = \log{z} = \ln{|z|}+i\arg{z}.$$
Here, the $\ln|z|$ means the usual Napierian or \PMlinkname{natural logarithm}{NaturalLogarithm2} (`{\em logarithmus naturalis}') of the real number $|z|$.\, If we fix the phase angle $\varphi$ of $|z|$ so that \,$0 \leqq \varphi < 2\pi$, we can write
$$\log{z} = \ln{r}+i\varphi+n\cdot 2\pi i\quad(n = 0,\,\pm1,\,\pm2,\,...).$$
The complex logarithm $\log{z}$ is defined for all \,$z \neq 0$\, and it is infinitely multivalued $-$ e.g.\, $\log{(-1)} = (2n+1)\pi i$\, where $n$ is an arbitrary integer.\, The values with\, $n = 0$\, are called the \PMlinkescapetext{{\em principal values}} of the \PMlinkescapetext{logarithm}; if $z$ is real, the \PMlinkescapetext{principal} value of $\log{z}$ coincides with $\ln{z}$.