# complex logarithm

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

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

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 natural logarithm (http://planetmath.org/NaturalLogarithm2) (‘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

 $\log{z}=\ln{r}+i\varphi+n\cdot 2\pi i\quad(n=0,\,\pm 1,\,\pm 2,\,...).$

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 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