PlanetMath: complex logarithm

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complex logarithm

(Definition)

The logarithm of a complex number is defined as every complex number 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 is an arbitrary integer. The values with are called the principal values of the logarithm; if is real, the principal value of $\log{z}$ coincides with $\ln{z}$ .

"complex logarithm" is owned by pahio.

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Other names:

natural logarithm

This object's parent.

Attachments:: general power (Definition) by pahio

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Cross-references: integer, angle, fix, real number, logarithmus naturalis, solution, equation, complex number
There are 12 references to this entry.

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 MSC:	30D20 (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)

Pending Errata and Addenda

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