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单词 GeneralPower
释义

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

zμ:=eμlogz=eμ(ln|z|+iargz).

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

zμ=eαln|z|-βargzei(βln|z|+αargz),

and thus

|zμ|=eαln|z|-βargz,argzμ=βln|z|+αargz.

This shows that both the modulus and the argument (http://planetmath.org/Complex) 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

|zμ|=|z|μ,argzμ=μargz.

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.

Examples

  1. 1.

    z1m, where m is a positive integer, coincides with the mth root (http://planetmath.org/CalculatingTheNthRootsOfAComplexNumber) 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 (http://planetmath.org/SequenceAccumulatingEverywhereIn11)).

  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.

Titlegeneral power
Canonical nameGeneralPower
Date of creation2013-03-22 14:43:17
Last modified on2013-03-22 14:43:17
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id31
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 30D30
Synonymcomplex power
Related topicLogarithm
Related topicExponentialOperation
Related topicGeneralizedBinomialCoefficients
Related topicPuiseuxSeries
Related topicPAdicExponentialAndPAdicLogarithm
Related topicFractionPower
Related topicSomeValuesCharacterisingI
Related topicUsingResidueTheoremNearBranchPoint
Definesbase of the power
Definesbase
Definesexponent
Definesbranch
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更新时间:2025/5/5 0:47:17