complex logarithm

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

$e^w=z.$ (1)

This is is denoted by

$$\log z:=w.$$

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

$$w=\log z=\ln |z|+i\arg z.$$

Here, the $\ln |z|$ means the usual Napierian or natural logarithm (http://planetmath.org/NaturalLogarithm2) (‘logarithmus naturalis’) of the real number $|z|$.  If we fix the phase angle $\phi$ of $|z|$ so that  , we can write

$$\log z=\ln r+i\phi +n\cdot 2\pi i\mathit{\hspace{1em}}(n=0,\pm 1,\pm 2,\mathrm{\dots }).$$

The complex logarithm $\log z$ is defined for all  $z\ne 0$  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 of the ; if $z$ is real, the value of $\log z$ coincides with $\ln z$.

Title	complex logarithm
Canonical name	ComplexLogarithm
Date of creation	2013-03-22 14:43:11
Last modified on	2013-03-22 14:43:11
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Definition
Classification	msc 32A05
Classification	msc 30D20
Synonym	natural logarithm
Related topic	Logarithm
Related topic	NaturalLogarithm2
Related topic	ValuesOfComplexCosine
Related topic	EqualityOfComplexNumbers
Related topic	SomeValuesCharacterisingI
Related topic	UsingResidueTheoremNearBranchPoint