complex logarithm

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

ew=z. (1)

This is is denoted by


The solution of (1) is obtained by using the form  ew=reiφ , where  r=|z|  and  φ=argz;  the result is


Here, the ln|z| means the usual Napierian or natural logarithmMathworldPlanetmathPlanetmath ( (‘logarithmus naturalis’) of the real number |z|.  If we fix the phase angle φ of |z| so that  0φ<2π, we can write


The complex logarithm logz is defined for all  z0  and it is infinitely multivalued - e.g.  log(-1)=(2n+1)πi  where n is an arbitrary integer.  The values with  n=0  are called the of the ; if z is real, the value of logz coincides with lnz.

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