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[parent] complex logarithm (Definition)

The logarithm of a complex number $ z$ is defined as every complex number $ w$ which satisfies the equation

$\displaystyle e^w = z.$ (1)

This is is denoted by
$\displaystyle \log{z} := w.$

The solution of (1) is obtained by using the form $ e^w = re^{i\varphi}$, where $ r = \vert z\vert$ and $ \varphi = \arg{z}$; the result is

$\displaystyle w = \log{z} = \ln{\vert z\vert}+i\arg{z}.$
Here, the $ \ln\vert z\vert$ means the usual Napierian or natural logarithm (`logarithmus naturalis') of the real number $ \vert z\vert$. If we fix the phase angle $ \varphi$ of $ \vert z\vert$ so that $ 0 \leqq \varphi < 2\pi$, we can write
$\displaystyle \log{z} = \ln{r}+i\varphi+n\cdot 2\pi i\quad(n = 0,\,\pm1,\,\pm2,\,...).$

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 principal values of the logarithm; if $ z$ is real, the principal value of $ \log{z}$ coincides with $ \ln{z}$.



"complex logarithm" is owned by pahio.
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See Also: logarithm, natural logarithm, values of complex cosine, equality of complex numbers

Other names:  natural logarithm

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general power (Definition) by pahio
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Cross-references: integer, angle, fix, real number, logarithmus naturalis, solution, equation, complex number
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This is version 7 of complex logarithm, born on 2004-10-10, modified 2008-01-25.
Object id is 6342, canonical name is ComplexLogarithm.
Accessed 9402 times total.

Classification:
AMS MSC30D20 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Entire functions, general theory)
 32A05 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Power series, series of functions)

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