general power

The general powerzμ, where z(0) and μ are arbitrary complex numbersMathworldPlanetmathPlanetmath, is defined via the complex exponential function and complex logarithm (denoted here by “log”) of the by setting


The number z is the base of the power zμ and μ is its exponent.

Splitting the exponent  μ=α+iβ  in its real and imaginary partsMathworldPlanetmath one obtains


and thus


This shows that both the modulus and the argument ( of the general power are in general multivalued.  The modulus is unique only if  β=0,  i.e. if the exponent  μ=α  is real; in this case we have


Let  β0.  If one lets the point z go round the origin anticlockwise, argz gets an addition 2π and hence the zμ has been multiplied by a having the modulus  e-2πβ1, and we may say that zμ has come to a new branch.


  1. 1.

    z1m, where m is a positive integer, coincides with the mth root ( of z.

  2. 2.

    32=e2log3=e2(ln3+2nπi)=9(e2πi)2n=9   n.

  3. 3.

    ii=eilogi=ei(ln1+π2i-2nπi)=e2nπ-π2   (with  n=0,±1,±2,);  all these values are positive real numbers, the simplest of them is  1eπ0.20788.

  4. 4.

    (-1)i=e(2n+1)π  (with  n=0,±1,±2,)  also are situated on the positive real axis.

  5. 5.

    (-1)2=e2log(-1)=e2i(π+2nπ)=ei(2n+1)π2   (with  n=0,±1,±2,);  all these are (meaning here that their imaginary parts are distinct from 0), situated on the circumference of the unit circleMathworldPlanetmath such that all points of the circumference are accumulation points of the sequence of the (-1)2 (see this entry (

  6. 6.

    21-i=2e2nπ(cosln2+isinln2)   (with  n=0,±1,±2,), are situated on the half line beginning from the origin with the argument ln20.69315 radians.

Title general power
Canonical name GeneralPower
Date of creation 2013-03-22 14:43:17
Last modified on 2013-03-22 14:43:17
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 31
Author pahio (2872)
Entry type Definition
Classification msc 30D30
Synonym complex power
Related topic Logarithm
Related topic ExponentialOperation
Related topic GeneralizedBinomialCoefficients
Related topic PuiseuxSeries
Related topic PAdicExponentialAndPAdicLogarithm
Related topic FractionPower
Related topic SomeValuesCharacterisingI
Related topic UsingResidueTheoremNearBranchPoint
Defines base of the power
Defines base
Defines exponent
Defines branch